Sankhara as a karmic-formation can be understood to be a schema, as defined by Immanuel Kant. From this point of view, sankhara as a cognitive process that transforms experience into belief. For example, by means of a sankhara process can take a set of sensations of a chair and "put them together" into a concept of a chair, in this way a person acquires perceptual knowledge about chairs. Kant called this a process of synthesis. Subsequent experiences with chairs can result in the growth or development of this perceptual knowledge, such as having feelings about and a desire for our favorite chair. The process of "putting together" is a form of mental action and therefore a form of kamma (karma in Sanskrit). The product of the process of "putting together" is "that which has been put together," which is also called sankhara. Defined in this way, sankhara or schemata are the cause of all intelligent behavior such as perception, feeling, believing, wishing, knowing, and having emotions. Such behavior can be wholesome, unwholesome, or functional. In advanced states of mindfulness meditation, it is possible to access the history of a karmic-formation and, if necessary, correct, revise, or unlearn it. I have written a book on how mindfulness meditation works that relates meditation to various sankhara.

And what are fabrications? These three are fabrications: bodily fabrications, verbal fabrications, mental fabrications. These are called fabrications.

"In-&-out breaths are bodily; these are things tied up with the body. That's why in-&-out breaths are bodily fabrications. Having first directed one's thoughts and made an evaluation, one then breaks out into speech. That's why directed thought & evaluation are verbal fabrications. Perceptions & feelings are mental; these are things tied up with the mind. That's why perceptions & feelings are mental fabrications."

‘Whatever fabrications , bhikkhus, has passed, ceased, changed: the term, label, and description ‘was’ applies to it, not the term ‘is’ or the term ‘will be.’

"And what are fabrications? These six classes of intention [pali cetana] — intention with regard to form, intention with regard to sound, intention with regard to smell, intention with regard to taste, intention with regard to tactile sensation, intention with regard to ideas: these are called fabrications. From the origination of contact comes the origination of fabrications. From the cessation of contact comes the cessation of fabrications. And just this noble eightfold path is the path of practice leading to the cessation of fabrications... The fact that pleasure & happiness arises in dependence on fabrications: that is the allure of fabrications. The fact that fabrications are inconstant, stressful, subject to change: that is the drawback of fabrications. The subduing of desire & passion for fabrications, the abandoning of desire & passion for fabrications: that is the escape from fabrications...

"Intention [cetana], I tell you, is kamma. Intending, one does kamma by way of body, speech, & intellect.

"And what is the cause by which kamma comes into play? Contact is the cause by which kamma comes into play.

"And what is the diversity in kamma? There is kamma to be experienced in hell, kamma to be experienced in the realm of common animals, kamma to be experienced in the realm of the hungry shades, kamma to be experienced in the human world, kamma to be experienced in the world of the devas. This is called the diversity in kamma.

"And what is the result of kamma? The result of kamma is of three sorts, I tell you: that which arises right here & now, that which arises later [in this lifetime], and that which arises following that. This is called the result of kamma.

"And what is the cessation of kamma? From the cessation of contact is the cessation of kamma; and just this noble eightfold path — right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration — is the path of practice leading to the cessation of kamma.

"And what is name-&-form? Feeling, perception, intention, contact, & attention: This is called name. The four great elements, and the form dependent on the four great elements: This is called form. This name & this form are called name-&-form.

“And why, bhikkhus, do you call them fabrications? ‘They fabricate the conditioned,’ bhikkhus, therefore they are called fabrications. SN 22.79

And what is the conditioned that they fabrications?

They fabricate conditioned form as form; they fabricate conditioned feeling as feeling; they fabricate conditioned perception as perception; they fabricate conditioned fabrications as fabrications; they fabricate conditioned consciousness as consciousness.

‘They fabricate the conditioned,’ bhikkhus, therefore they are called fabrications.

(Here end sutta excerpts)

As ChrisW says “The word sankhara is derived from the prefix sam, meaning "together," joined to the noun kara, "doing, making."

saṅkhāra = come together & do.

What precisely does “come together” mean?

It means “orienting oneself” in sensory experience.

It means saying “I am here”. (See the 36 streams of craving).

saṅkhāra = get oriented & do.

saṅkhāra = in THIS place with THIS aim, DO this

saṅkhāra = a sensory motor prediction

The sensory motor brain predicts sensory experience.

We construct maps to guide our movements in the world.

All maps, including the dhamma, are composed of saṅkhāra.

Each saṅkhāra starts off as a hypothesis with a degree of uncertainty.

However, as it predicts sensory experience correctly, it hardens into a “habit” and we cease to be conscious of the fact that it is just a hypothesis and may be proved incorrect by future sensory experience.

What does this have to do with “suffering & the end of suffering”?

Āsava are saṅkhāra corrupted by avijjā.

Āsava are saṅkhāra which predict sensory experience badly.

The destruction of the āsava is the end of suffering.

Why?

Because dukkha is a notification of misprediction.

The biological purpose of dukkha is to alert us to an error in our predictive model of the world of sensory experience.

How we respond to this arrow in the heart is important.

If we respond by ignoring the feedback and clinging to our wrong map of the world, we create an inner civil war between (1) the part which wants to predict sensory experience accurately so that our movements will be skillful and (2) the part which wants to cling to the wrong view.

This inner civil war between these 2 parts is all suffering.

All suffering that we have ever experienced is an inner civil war between these 2 parts.

When, with viriya, this bad habit of avijjā [ignoring uncomfortable truths] (avijjāsava) is extinguished, all suffering is ended.

Most in depth explanations happens in the Abhi Dhamma. There are 52 Cetasikas our of which 50 come under Sankara (with the exception of Feeling and Perception). Explaining the 50 would need a small size book and maybe beyond the scope of an answer in this format hence refer to: Mental Factors By Dharmacharya Ruwan Buddhika, The Abhidhamma in Practice by N.K.G. Mendis, Comprehensive Manual of Abhidhamma, Mental factors.

According to the Suttas there are 3 types of Sankaran (Cūla Vedalla Sutta, (Paṭicca,samuppada) Vibhanga Sutta):

1. the bodily formations - breath
2. the verbal formations - thinking and pondering
3. the mental formations - feeling and perception

2 above related to the 50 Cetasikas in the Abhi Dhamma while 3 refers to the other 2 Cetasikas. Mental formations are the last to calm. (Anupubba Nirodha Sutta)

Sankara are formed when there is either greed (lobha), hatred (dosa), delusion (moha), greedlessness (alobha), hatelessness (adosa), and undeludedness (amoha) volitional motivated thoughts (this also based on sensation as any through has sensation associated with it) or when you react with clinging or aversion to any sensation that is felt. (Pahāna Sutta, Avijja Pahana Sutta 2)

I recommend the book "How to Measure and deepen your spiritual realization" for the most modern analysis and cross correlations. It has been 15 years since i read this and i have not found anything better. There is a 100+ page chapter to analyzing the skandhas (including sankhara) and how they relate to daily life, samadhi, and the various gongfus as described in the surangama sutra (which is a Mahayana sutra, by the way), various modern Buddhist books, and some Theravada references as well since the 5 aggregates is a universal dharma common to many schools.

I think this entire universe is made up of Sankara, which cannot be seen thorough our 5 senses(Vinyana-Perception).Also "San" means collection and "kara" means information. It is the collection of information. Buddha said "Pragnya" eye can see this small particles..Buddha told Rahula, "you and that tree are the same"...it has a nature of "create" and "cessation" at the same time...

Sankhara (samskara, sanskara) is a technical term for assembled phenomena. Why are they called "assembled"?

Phenomena are assembled in at least three senses:

1. They are assembled from parts.
2. They are assembled by the coming together of direct causes and ambient conditions necessary for their arising and preventing their disintegration.
3. They are assembled by the mind providing a context/perspective and delineating them from other adjacent phenomena, in accordance with some organizational principle.

Sankhara-khandha (samskara-skandha) is a functional group providing material for assembling of phenomena. It is made of imprints left by previous experiences, known as vasanas - elementary memories or associations.

The canonical sayings about Sankhara include (the idiomatic translation is mine):

• anicca vata sankhara, uppada-vaya-dhammino (impermanent are, all assembled phenomena - their nature is emergence and dissipation)
• sabbe sankhara dukkha (all assembled phenomena are faulty)
• vayadhamma sankhara, appamadena sampadetha (liable to dissipation are all assembled phenomena, through disenchantment shall succeed)

A synonym of Sankhara is "Samudaya", a word that is often translated as "origination" or "arising" (as in dukkha samudaya - "origination of suffering", the Second Noble Truth) but whose literal meaning is "coming together", "combination", "aggregation", "collection", "junction", "cluster", "union". Hence the famous maxim known as The Eye of The Dharma:

yaṃ kiñci samudayadhammaṃ, sabbaṃ taṃ nirodhadhamman'ti (whatever is assembled -- all that can be prevented)

The Pali word ‘sankhara’ has multiple meanings dependent on the context. It can mean:

(i) a conditioned thing (formed from various causes or parts);

(ii) something that conditions another conditioned thing, i.e., a ‘conditioner’;

(iii) the process of conditioning;

(iv) the aggregate of mental forming, such as desiring, intention, thinking, etc and

(v) mental conditioning/concocting by greed, hatred & delusion, i.e., uncontrolled disturbing mental proliferating.

For example, shampoo is a ‘conditioned thing’, because it is manufactured from & comprises of many ingredients. When shampoo is applied to hair, a ‘conditioning’ process takes place, which makes hair soft, clean & shiny. Therefore, shampoo is also a (hair) ‘conditioner’.

Examples of each context from the Pali suttas are below:

1. ‘Sankhara’ as ‘conditioned things’.

Rūpaṃ kho āvuso channa, aniccaṃ, vedanā aniccā, saññā aniccā, saṃkhārā aniccā, viññāṇaṃ aniccaṃ, rūpaṃ anantā, vedanā anattā, saññā anattā, saṃkhārā anattā, viññāṇaṃ anattā, sabbe saṃkhārā aniccā, sabbe dhammā anattā

Form, friend Channa, is impermanent. Feeling is impermanent. Perception is impermanent. Mental formations (saṃkhārā) are impermanent. Consciousness is impermanent. Form is not-self. Feeling is not-self. Perception is not-self. Mental formations are not-self. Consciousness is not-self. All (sabbe) conditioned things (saṃkhārā) are impermanent (aniccā). All (sabbe) phenomena (dhammā) are not-self (anattā).

SN 22.90

The verse above contains the word ‘sankhara’ twice: (i) as the aggregate of mental forming; and (ii) as all conditioned things, referring to each of the aggregates.

This stock phrase – sabbe saṃkhārā aniccā – is found in the exact same context at AN 3.134 and at Dhammapada 277.

The five aggregates are conditioned things because they arise & exist subject to & dependent upon various causes and conditions. Thus, SN 22.21 states:

Rūpaṃ kho ānanda, aniccaṃ, saṃkhataṃ paṭiccasamuppannaṃ khayadhammaṃ

Form…feeling…perception…mental formations…consciousness are impermanent, conditioned (saṃkhataṃ), dependently arisen (paṭiccasamuppannaṃ), subject to destruction (khayadhammaṃ)…

Therefore, in this context, ‘sankhara’ also refers to physical or material things, as found in the Pali terms ‘kaya sankhara’ (which is defined in MN 44 as the ‘in & out breathing’) & ‘aayu sankhara’ (defined in MN 43 as the ‘vitality’ or ‘life force’ of the physical body that remains when the mind is unconscious).

2. ‘Sankhara’ as ‘conditioners’.

Here, ‘sankhara’ as ‘conditioners’ refers to the words ‘kaya sankhara’, ‘vaci sankhara’ & ‘citta sankhara’, which are found at the 2nd condition of Dependent Origination & at stages 4, 7 and 8 of the Anapanasati Sutta and which are defined in MN 44 as follows:

In-&-out breaths are the body conditioner (kāyasaṅkhāro). Initial & sustained thought are the verbal conditioner (vacīsaṅkhāro). Perceptions & feelings are the mind conditioner (cittasaṅkhāroti).

Why are in-&-out breaths the body conditioner? Why are directed thought & evaluation the verbal conditioner? Why are perceptions & feelings the mind conditioner?

In-&-out breaths are a bodily thing upon which the body depends on (or is bound on). That’s why in-&-out breaths are the body conditioner. Having first applied & sustained one’s thought, one then breaks out into speech. That’s why applied & sustained thought are the verbal conditioner. Perceptions & feelings are mental; these are things that determine mental states (i.e., upon which mental states depend). That’s why perceptions & feelings are the mind conditioner.

The above translation has been done liberally to demonstrate the meaning. However, the part of the translation about why thought is the verbal conditioner is the same as all translators. This part demonstrates clearly what the mean of ‘sankhara’ is in this context, namely, the ‘verbal sankhara’ is a preceding cause that determines the happening of speech. Thus ‘thought’ is not the ‘verbal condition’ or ‘verbal fabrication’ but, instead, the ‘verbal conditioner’ or the ‘verbal fabricator’.

Thus, the ‘kaya’ & ‘cittta sankhara’ have the same meaning. The breathing conditions the state of the body. Breathing gives life to the body. If breathing is healthy, the body is healthy & energetic. If breathing is poor, the body is unhealthy & weak. This is why breathing is the body ‘conditioner’ (rather than the bodily condition). Similarly, pleasant feelings & perceptions condition mental formations of love, greed or lust; whilst unpleasant feelings & perceptions condition mental formations of anger or hatred. This is why perception & feeling is the mind ‘conditioner’ (rather than the mental condition).

3. ‘Sankhara’ as ‘conditioning’.

Below are two straightforward examples of the word ‘abhisaṃkharontīti’.

Kiñca bhikkhave, saṃkhāre vadetha: saṃkhataṃ abhisaṃkharontīti bhikkhave

And why do you call them ‘fabricators’? Because they fabricate fabricated things, thus they are called ‘fabricators.’

SN 22.79

Avijjāgatoyaṃ bhikkhave, purisapuggalo puññaṃ ce saṅkhāraṃ abhisaṅkharoti

Monks, if a person immersed in ignorance generates a meritorious formation…

SN 12.51

4. ‘Sankhara’ as the ‘mental formations aggregate’.

There are five aggregates (khandha) of life (body, feeling, perception, mental formations & consciousness), of which ‘sankhara khandha’ (mental formations/forming) is the 4th.

The ‘sankhara khandha’ operates in all people, including Buddhas. For example, about the end of the life of a fully enlightened being, the scriptures state:

Then, friend Yamaka, how would you answer if you are thus asked: A monk, a worthy one, with no more mental effluents: what is he on the break-up of the body, after death?

Thus asked, I would answer, ‘Form is impermanent… Feeling… Perception… Fabrications… Consciousness is impermanent. That which is impermanent is unsatisfactory. That which is unsatisfactory has ceased and gone to its end.’

Very good, my friend Yamaka. Very good.

Yamaka Sutta

Therefore, for a Buddha to think or speak, the sankhara aggregate must operate in the mind/brain of a Buddha. In this context, ‘sankhara’ is not ‘mental suffering’ but referring to a neutral mental faculty.

5. ‘Sankhara’ as the ‘mental concocting’.

There are many passages about how ‘sankhara’ is the greatest suffering & about how Nibbana is the stilling of ‘sankhara’. Since a Buddha must think (in order to teach) & is comprised of five undefiled pure five aggregates, the term ‘sankhara’ is these passages does not refer to ‘sankhara’ as ‘conditioned things’ or the ‘sankhara aggregate’. Sankhara, here, also probably does not refer to ‘sankhara’ as ‘conditioners’ since Buddhas still have breathing, feeling, perception & thought. Therefore, ‘sankhkara’ here must refer to the mind being ‘conditioned’, ‘concocted’, ‘stirred up’ & ‘polluted’ by greed, hatred & delusion, i.e., uncontrolled disturbing mental proliferating.

Some relevant well-known passages are below:

Saṅkhāra paramā dukhā…nibbāṇa paramaṃ sukhaṃ.

Mental concocting is the worst suffering….Nibbana is the highest bliss.

Dhammapada 203

Tisso imā āvuso dukkhatā, dukkhadukkhatā saṅkhāradukkhatā vipariṇāmadukkhatā.

There are these three kinds of suffering, my friend: suffering due to/about pain; suffering due to/of mental concocting; and suffering due to/about change. These are the three kinds of suffering.

SN 38.14

Construing is a disease, construing is a cancer, construing is an arrow. By going beyond all construing, he is said to be a sage at peace.

MN 140

Idampi kho ṭhānaṃ duddasaṃ yadidaṃ sabbasaṅkhārasamatho sabbūpadhipaṭinissaggo taṇhakkhayo virāgo nirodho nibbānaṃ.

This state, too, is hard to see: the calming (samatho) of all (sabba) mental concocting (saṅkhāra), the relinquishment of all acquisitions, the destruction of craving; dispassion; cessation [of suffering]; Nibbana.

MN 26

Conclusion: In the Pali, the word ‘sankhara’ is used in many different contexts. For a clear & practical comprehension of the Pali teachings, the suitable context is best discerned.

Below is an example of an internet translation & the alternate translation of this webpage.

These three feelings have been spoken of by me: a feeling of pleasure, a feeling of pain & a feeling of neither pleasure nor pain. These are the three feelings spoken of by me. But I have also said: ‘Whatever is felt comes under stress.’ That I have stated simply in connection with the inconstancy of fabrications (sankhara). That I have stated simply in connection with the nature of fabrications to end… in connection with the nature of fabrications to fall away… to fade away… to cease… in connection with the nature of fabrications to change.

These three feelings have been spoken of by me: a feeling of pleasure, a feeling of pain & a feeling of neither pleasure nor pain. These are the three feelings spoken of by me. But I have also said: ‘Whatever is felt comes under unsatisfactoriness.’ That I have stated simply in connection with the inconstancy of conditioned things (sankhara). That I have stated simply in connection with the nature of conditioned things (sankhara) to end… in connection with the nature of conditioned things to fall away… to fade away… to cease… in connection with the nature of conditioned things to change.

SN 36.11

There's an article in English titled Anicca Vata Sankhara in which Bhikkhu Bodhi describes the various meanings of the word. Here are very brief extracts from that article, a summary:

The word sankhara is derived from the prefix sam, meaning "together," joined to the noun kara, "doing, making."

1. The suttas distinguish the sankharas active in dependent origination into three types: bodily, verbal, and mental.
2. A second major domain where the word sankharas applies is among the five aggregates. The fourth aggregate is the sankhara-khandha, the aggregate of volitional formations. The texts define the sankhara-khandha as the six classes of volition (cha cetanakaya): volition regarding forms, sounds, smells, tastes, tactile objects, and ideas.
3. The third major domain in which the word sankhara occurs is as a designation for all conditioned things. In this context the word has a passive derivation, denoting whatever is formed by a combination of conditions; whatever is conditioned, constructed, or compounded.

The suttas

I don't know whether there is a good/long/detailed canonical description in the Suttas. Suminda's answer references the Abhidhamma.

One of the places it's described is the Paticca-samuppada-vibhanga Sutta, very short:

And what are fabrications? These three are fabrications: bodily fabrications, verbal fabrications, mental fabrications. These are called fabrications.

The aggregates

This page contain three essays by Ajaan Lee. The last essay on that page titled "Part III: The Buddhist Way" describes various types of Sankhara, which is "a brief outline of the Buddha's teachings based on the synopsis of the Ovada Patimokkha". It includes (again I'm heavily summarizing the article):

• "Sankharas on the level of the world": status, fortune, praise
• "Sankharas on the level of the Dhamma": Dhatu, Khandha, and Ayatana

I think that all khandhas are sankharas, i.e. each khandha is some-things-put-together; for example they include:

• Putting things together so that they can be sensed (or more specifically, contact between the sense and sense-object)
• Putting a "feeling" with the thing (e.g. "I feel that's good" or "I think that's bad")

The sense of self (identity view) is presumably a sankhara too (see the simile of the chariot).

The same article also mentions some important attributes of sankharas, e.g. impermanence:

The Buddha taught in line with the true nature of the world. He said, "Khaya-vaya-dhamma sankhara, appamadena sampadetha," which means, "All sankharas, once they have arisen, decay by their very nature. Don't be heedless or complacent. Be thoroughly mindful and completely alert, and you will attain peace and security."

What this means is this: All things that appear in the world arising from actions (kamma) are called sankharas — formations, fashionings, compounded things. Sankharas, by their nature, or of two sorts — sankharas on the level of the world and sankharas on the level of the Dhamma.

I may be wrong but I suspect that vipassana meditation might be supposed to help you understand that, at least at the sensory level. I think vipassana is supposed to make you aware of impermanence, but because sankharas are impermanent perhaps "becoming aware of impermanence" means "becoming aware of sankharas".

I also like the explanation, "All things that appear in the world arising from actions (kamma) are called sankharas": because I think that helps to explain why sankharas are sometimes translated as not just "formations" but "volitional formations": i.e. perhaps it's because kamma (action) is also associated with will (see "This volition and the resultant actions constitute kamma" in this commentary).

One of the five aggregates

The terminology turns out to be confusing because all five khandhas are sankharas, but also one of the khandhas in particular is called sankharas.

I think that means "all khandhas are fabricated or put together, but one of the khandhas is what people do when they put things together or fabricate things in their mind."

Wikipedia summarizes this meaning of sankhara as a specific one one of the five aggregates:

4. "mental formations", "impulses", "volition", "fabrications" or "compositional factors" (Skt. samskāra, Pāli saṅkhāra, Tib. 'du-byed): all types of mental habits, thoughts, ideas, opinions, prejudices, compulsions, and decisions triggered by an object

This too matches (is an even better match for) the "volitional formation" translation.

The footnote to that Wikipedia paragraph begins with, "The Theravada Abhidhamma divides saṅkhāra into fifty mental factors".

All conditioned things

There's an example of this usage in the Buddha's last words (in the The Mahāparinibbāna Sutta),

8. And the Blessed One addressed the bhikkhus, saying: “Behold now, bhikkhus, I exhort you: All compounded things are subject to vanish. Strive with earnestness!”⁵⁸

This was the last word of the Tathāgata.

The footnote gives the Pali,

Handa dāni bhikkhave āmantayāmi vo: Vayadhammā saṅkhārā appamādena sampādetha.

Jayarava explores that phrase, and how difficult it is to translate, in this article, The last words of the Buddha.