What is Prana according to Buddhism?

In Theravada Buddhism, we call it pāṇa, which is just the Pali cognate of the Sanskrit prāṇa. The word itself just means breath, though it has connotations that come from Hinduism, regarding the concepts of life essence, soul, energy, etc. The PED gives some idea of how it is used in Buddhism:

Pāṇa [fr. pa+an, cp. Vedic prāṇa breath of life; P. apāna, etc.] living being, life, creature D iii.48, 63, 133 S i.209, 224; v.43, 227, 441 (mahā -- samudde); A i.161 ii.73, 176, 192; Sn 117, 247, 394, 704; Dh 246; DA i.69 161; KhA 26; ThA 253; PvA 9, 28, 35; VvA 72 DhA ii.19. -- pl. also pāṇāni, e. g. Sn 117; Dh 270. <-> Bdhgh's defⁿ of pāṇa is "pāṇanatāya pāṇā; assāsapassās' āyatta -- vuttitāyā ti attho" Vism 310.

-- âtipāta destruction of life, murder Vin i.83 (in "dasa sikkhāpadāni," see also sīla), 85, 193; D iii.68, 70, 149 182, 235; M i.361; iii.23; Sn 242; It 63; J iii.181 Pug 39 sq.; Nett 27; VbhA 383 (var. degrees of murder) DhA ii.19; iii.355; DA i.69; PvA 27. -- âtipātin one who takes the life of a living being, destroying life D iii.82; M iii.22; S ii.167; It 92; DhA ii.19. -- upeta possessed or endowed with life, alive [cp. BSk. prāṇopeta Divy 72, 462 etc.] S i.173; Sn 157; DA i.236 -- ghāta slaying life, killing, murder DA i.69; -- ghātin âtipātin DhA ii.19. -- bhu a living being J iv.494 -- bhūta=˚bhu M iii.5; A ii.210; iii.92; iv.249 sq. J iv.498. -- vadha=âtipāta DA i.69. -- sama equal to or as dear as life J ii.343; Dpvs xi.26; DhA i.5. -- hara taking away life, destructive M i.10=iii.97; S iv.206 A ii.116, 143, 153; iii.163.

In Theravada Buddhism this is as far as the concept goes, and it is only conceptual; there is no sense of an actual "spirit" or "energy" living inside of the body that can be called pāṇa; it is only the concept of a living, breathing being that is described as pāṇa. As the Visuddhimagga explains:

Breathing things (pāṇa): so called because of their state of breathing (pāṇanatā); the meaning is, because their existence depends on in-breaths and out-breaths.

Vism. IX.54 (Nyanamoli, trans)

In Tibetan Buddhism, though, there seems to be more of a recognition of the existence of an energy flowing through the body, a "psychic wind" (Tibetan: rLung), so to speak:

Some of the different usages of the term lung include:

• the psychic winds (sanskrit: prana) that travel in the internal channels, or nadi (Sanskrit) of the subtle body and are manipulated in certain Vajrayana yoga practices.
• specifically the five psychic winds that are a manifestation of the Mahabhuta. These five are the lifeforce that animate the bodymind (Sanskrit: namarupa) of all sentient beings and are key to certain tantric Buddhist and Bon sadhanas and Traditional Tibetan medicine.
• to the vayu and prana of ayurvedic medicine.
• as a component of the term for a type of prayer flag, named after the allegorical Wind Horse (Tibet: lung ta).
• a type of tantric buddhist empowerment that involves the transference of spiritual power from master to augment or refine that of the disciple through the recitation of scripture or song. This oracular transmission received aurally defines Mantrayana and Ngagpa traditions and provides them with their nomenclature.
• the "reading transmission" of sutrayana texts, in which the entirety of the text is read aloud from teacher to student.

-- Wikipedia, Lung (Tibetan Buddhism)

Tenzin Wangyal Rinpoche devotes an entire section of his book "The Tibetan Yogas of Dream and Sleep" to the topic of prana.

In answer to your auxiliary questions, it again depends somewhat on which tradition you follow, but I think the consensus is that only sentient beings share the quality of prana, however it is understood.

While I've hardly ever heard the word "prana" used in context of Buddhism proper (only in context of Yoga), there are several concepts in some Buddhist traditions that seem to refer to the same underlying principle.

Chinese, Korean and Japanese Buddhists speak about "qi" or "ki". Tibetan Buddhists speak about "lung". Chogyam Trungpa Rinpoche called it "abstract energy"...

It's kind of similar to the notion of "potential energy" in physics, but applied to the subjective realm of organic sentient beings. And it's much broader, so it includes the bodily/physical energy as well as emotional energy and even information -- in their aspect of latent potentiality.

So if you are physically sick or tired, we say your energy is low. If you are emotionally down, we also say your energy is low. And why are you emotionally low? Could be because of the physical reasons -- or could be because of the information you've exposed yourself to, recently! So, if you lack confidence, or you lack drive -- we also say your energy is low. Now, if your teacher helps you get out of depression and build self-confidence, by giving you certain information -- we say he gave you some of his energy. You now run on your teacher's energy. Then, if his advice did not produce a permanent change in you, you may again fall into depression -- and then we say you have exhausted the energy teacher gave you.

Similarly we say that coffee and tea have energy -- because of their potential ability to affect our condition. Certain plants have more energy than others. At this point we understand that there are multiple kinds of energy (tamas, rajas, sattva) -- but all these are just concepts, just ways to describe different aspects of the way the world works. None of this exists as some kind of electromagnetic field -- it is only an abstraction, just like the potential energy in physics.

If we go further along, we can see that books may also transmit energy. Not only books, but also some objects. Some places could evoke energy, right? So where does this energy come from, where does it belong, where does it originate? It is just a part of the world, passing across, interweaving, transforming continuously.

But generally speaking, this concept of energy is rooted in the experiential realm of living sentient beings. So when we speak about early morning air having lots of prana, it just means it has a certain atmosphere that affects our energy in a very specific way. So if we had, say, a computer running an AI program, even if it was equivalent to a human as far as intelligence -- it would not be able to appreciate the early morning prana, because it is not affected the same way as us by the morning mood. In this sense we could say that energy only exists inside of sentient beings.

So is energy a subjective phenomena of sentient beings, or is it objectively a part of the universe? At the end of the day, we hit the limits of what dualistic mind can explain and understand. There is no strict boundary between sentient beings and the world. There is no strict boundary between "inner" and "outer". The subjective and the objective are two sides of the same coin. The world is one continuous field of matter/energy/information, fluctuating.

I know this is not exactly scientific, but it is very difficult to speak about these things in an organized manner, because they are so touchy/feely and subjective. I hope this helps someone to overcome their doubts and discover the wonderful world of energy.

This is somewhat related to Kaya-sankara, i.e., the breath which sustains life. This might differ from what the Hindu concept as Buddhism does not have a soul concept.