What exactly is svabhava?

The ideas of 'svabhāva' in Nagarjuna's Madhyamaka are merely that, namely, "ideas" or intellectual thoughts rather any direct insight from meditation.

For example, based in meditation & insight, consciousness is mere awareness or consciousness and meditation & insight cannot reduce it to any smaller parts or particles (although meditation can discern the impermanence, selfless & how consciousness is dependent on sense organs to arise).

The following Pali teaching is an example of what 'sabhava' is meant to mean, namely, natural things have their own nature that is not a 'self' or 'mine':

And why, bhikkhus, do you call it form? ‘It is deformed,’ bhikkhus, therefore it is called form. Deformed by what? Deformed by cold, deformed by heat, deformed by hunger, deformed by thirst, deformed by contact with flies, mosquitoes, wind, sun, and serpents. ‘It is deformed,’ bhikkhus, therefore it is called form.

And why, bhikkhus, do you call it feeling? ‘It feels,’ bhikkhus, therefore it is called feeling.

And why, bhikkhus, do you call it perception? ‘It perceives,’ bhikkhus, therefore it is called perception.

And why, bhikkhus, do you call them formations? ‘They construct the conditioned,’ bhikkhus, therefore they are called formations.

And why, bhikkhus, do you call it consciousness? ‘It cognizes, ’ bhikkhus, therefore it is called consciousness.

SN 22.79

While the only term in the suttas that comes near to 'sabhava' is 'sadhatu', the meaning of 'sabhava' is 'inherent nature' rather than 'inherent self-existence'. The following sutta is about the inherent nature or 'sabhava' or 'sadhatu' of the Dhamma Dhatu (Element).

Uppādā vā bhikkhave tathāgatānaṃ anuppādā vā tathāgatānaṃ ṭhitāva sā dhātu dhammaṭṭhitatā dhammaniyāmatā

Monks, whether or not there is the arising of Tathagatas, this property stands — this steadfastness of the Dhamma, this orderliness of the Dhamma: All processes are inconstant. All processes are unsatisfactory. All phenomena are not-self.

Dhamma-niyama Sutta

What exactly is svabhāva in Nagarjuna's Madhyamaka?

I believe that in some abhidharma it is a dharma which makes other dharmas have a time. So without it, they may have argued, nothing could exist, so "everything exists". Take this from the Sarvastivada page

The Sarvāstivāda (Sanskrit; Chinese: 說一切有部; pinyin: Shuō Yīqièyǒu Bù) were an early school of Buddhism that held to the existence of all dharmas in the past, present and future, the "three times".

The abhidharma in the Taisho is I believe the Sarvastivada one (all the early schools had different versions).

I very much like this interpretation: Nagarjuna just points out that the assumption that anything has to exist in different times is bunk.

However, much Mahayana Buddhism uses the term in a different way, to mean what the scholar Westerhoff calls "essence svabhava". I believe I have it on very good (academic) authority that's what Zhiyi means when he talks about emptiness of svabhava. I can't really guess if that's from genuine, not merely contingent / historical, evolution of the term, as it features in Buddhist "philosophy". But if I may I'll add my own opinion: Zhiyi and his tientai school in general are talking about the exact same concept but refined it.

Wetsterhoff claims

Nagarjuna's Madhyamaka: A Philosophical Introduction. By Jan Westerhoff

So, if right, then Buddhists were able to limit, narrow down, the meaning of 'svabhava'.

We use this word very commonly in India.

Swabhava means nature of the self. For example if I ask what is your swabhav ? then I am asking whether you are loving , kind ,joyful ,compassionate, rude or stubborn kind of person? What kind of person you are ? It is personality trait.There are wholesome swabhavs which make you say that a person is very nice.There are unwholsome swabhavs which are not good.

Swabhav can change. For example, I was a very careless person in the past but now I am not that careless. You can use the word with any living being whether it is dog or cat or human. Every being develops a swabhav or inherits a swabhav from the previous birth.

If we break the word swabhav then "swa" means self and "bhav" means nature,i.e. nature of the self. Swabhav is not permanent and unchangeable therefore no swabhav is worth identifying as self.

Does it carry the same meaning as being unconditioned?

No.

Why does a chair or a photon or empty space or concepts or the Dharma (teachings) or Nirvana have no svabhāva?

Strictly speaking swabhav is associated with self. You don't say chair has a bad swabhav. Or the photon or empty space has good swabhav.

My swabhav or your swabhav has been evolving and will continue to evolve but there is no permanent nature which you can call as self. Swabhav falls in the category of five aggregates as mental formations or sankharas.

In abhidhamma:

Sabhāva = saka-bhava = own saṇkhāra = saṅkhata = sappaccaya = sahetuka = an effect of causes.

So, the answer is "an effect of causes".

Note: This is the same reason that why saṅkhāra equal to kamma-bhava.

Svabhāva, traditional Chinese Buddhist text translated as "自性", i.e., Self-nature. A Svabhava is a Thing-of-itself, it exists on it's own, cannot be deconstructed by the Eight Conditions quested by Nagarjuna; it is not caused by Causal-correlations (因緣), nor Compounded (和合). However, the world is made by (1)Causal-correlations and (2)Compounding, of every phenomena that the intellectual mind able to conceive. Therefore, it is empty. Because anything that's coming from Causal-correlations or Compounding, when investigating by the intellectual examination incisively with a superior mind, the result is - unobtainable. For nothing is obtainable therefore, the term to address it: "Emptiness".

My elaboration is, Causal-correlations and Compounding shorted as Correlative-arising (緣起). In my investigation, "dependent-arising", or most of the English translated terms I came across are, questionable; creating unnecessary hindrances to understand what precisely it meant.

(1) Causal-correlations

There are four types of Causal-correlations, Sanskrit: Catvārah-pratyayāh (四緣). Nagarjuna listed and deconstructed each as:

_{(the translated terms are my designations)}

a) Correlative-cause (因緣): Cause and effect. Deconstructed as: an effect not appeared the cause can't be defined; if the effect not in the cause, the cause cannot cause the effect, but if the effect already in the cause the effect doesn't need the cause for it existed already... etc.

b) Consecutive-cause (次第緣): A temporal cause, i.e., with the factor of time. Deconstructed as: if the effect not appeared, the cause can't disappear; however if the cause not disappeared how could the effect appear... etc.?

c) Causative-cause (緣緣): A spatial cause, i.e., with the factor of another object. Deconstructed as: for due to a) the cause and effect cannot be established therefore not a "thing" there, there's no object to give as a cause.

d) Superlative-cause(增上緣): All the above causes multiplied. Deconstructed as: since the above three can't be established this also null.

I'm afraid Jay Garfield's understanding of the Catvārah-pratyayāh which a very basic Abhidharma concept so questionable in his The Fundamental Wisdom of The Middle Way; I read his chapter 10, then 1 and 2, put aside.

(2) Compounding Compounding is unobtainable. For if things are different, they can't be compounded to be one thing; yet if they are the same, can't be compounded either, for same thing doesn't compound :). (I was laughing all the way when reading Madhyamaka (Classical Chinese translated by Kumarajiva) the 1st time, now I can't help laughing again when writing at here :))... difference due to differing therefore differences, different object unobtainable... etc.

Since all phenomena of the world are made by Causal-correlations and compounding, none is having Svabhava. Since all phenomena none is with Self-nature therefore it is empty, i.e., unobtainable (不可得). It is impermanent, conditioned. However, Nagarjuna didn't stop at here, not stopping at deconstructing, what he pointed to is, what beyond this Emptiness that given rise to all phenomena, the world and sentient... etc. Here is most of the scholastic Buddhist Masters/teachers fell. Such as, Tibetan Tsongkhapa by holding onto Madhyamika-Prasangika (wrong) viewpoint constructed his two Chenmo(s) misleading many Buddhists, bolting on the Intellect-mind (意識心), the 6th Vijñāna as permanent [which needed maintenance of each day 16 hours of xxx Tantric O_gsM, all months, all years... well maybe good news for some, who knows!? ;] .

Nirvana and the Buddhahood are of the 5th Statement. The Intellect-mind is able to come as far as the Four-statements (Catuskoti) limitation. To go beyond the Intellect-mind, what should be looked for by all practitioners but the Self-nature as in Ch'an (6th Patriarch Huineng: How so the Self-nature, originally is self-sufficient (何期自性, 本自具足)), or the Tathagatagarbha, or the Alaya... the teachings only available in the Mahayana Canon (excluding the later Tantric texts). One of the most important Sutras is Prajna-paramita Sutra, which the related is Heart Sutra. I'm afraid Thich Nhat Hanh though a respectable teacher I respected, his interpretation of Heart Sutra as quoted in this post is questionable. It's only max. to the level of a Śrāvaka, not for the Bodhisattva or Buddha, i.e., not of the Prajna-paramita Heart Sutra the ultimate Emptiness.

The OP asked another question on usefulness of Madhyamaka, the key answer is, due to the decline of man's capacity, very rare few able to see directly the Ultimate Reality (still an ignorant claims able to see in meditation and, libeling Nagarjuna :), as stated in the beginning of the Madhyamaka Chinese Text. One of the purposes of Nagarjuna, I think, was enabling a glimpse of Ultimate Reality by pushing the Intellect-mind to the ultimate, by exhausting the Four-statements. If one able to read the original Classical Chinese one will at least sharpen his mind, have a shot of intelligence. Those who incessantly libeling the Mahayana Sutras, pity are they misled by their teachers or traditions. However, this very act is so detrimental to wisdom cultivation even to the worst severed the root of Prajna-paramita rendered these people the Icchantikas warned the Buddha. The Buddha explaint what is Icchantika and why they became the Icchantikas, one of the karmic deeds was they libeled Mahayana teaching due to their ignorance they lack the wisdom to understand the profound deep deep Mahayana doctrines, recorded in Prajna Paramita Sutra. There some so-called "Sutras" inserted, the later Tantric text, unfortunately, in the Tripitaka; but those great Sutras of Prajna-paramita - The 2nd Turning, also Tathagatagarbha - The 3rd Turning, are spoken by Buddha Shakyamuni.

Svabhāva stands for inherent existence.

From an early Madhyamika viewpoint, there are two types of dependent-arising:

1. Causal dependence
2. Dependence on parts

While impermanent phenomena depend on both causes and parts, permanent phenomena do not depend on causes and conditions but they do depend on parts. For instance, a chair is impermanent and depends on causes (if it is made of wood, wood is a substantial cause, etc) and depends on its parts (the legs, etc). Permanent phenomena such as space, emptiness, cessations, etc. depend on parts. For instance, emptiness is emptiness of something on which it depends (not causally, but in terms of part). Therefore, even emptiness, a permanent phenomena, is empty of inherent existence because it is a dependent-arising.

From a Madhyamika-Prasangika viewpoint, there are two types of dependent-arising. The third level is the most subtle:

3. Dependence on name.

We say that the meaning of something being empty of svabhāva is that, if you look for it with a mind analyzing the ultimate, you will not find it. Now, if you ask yourself "is the chair its legs... is it the material it's made of... etc" you will find that the chair is merely labeled upon these without being any of these. We say that it is "merely labeled".

This presentation is incomplete and cartoonish because there are various Madhyamika [sub]schools, some for which inherent existence and true existence are synonymous, some not.