Centaurs, of course, originally appear in the myths of Ancient Greece, from whence they were borrowed by the Romans, and, later still, often seen in medieval bestiaries. (Similar creatures do appear in some other mythologies, but the true centaur that we're talking about here is the Greek one). They have regularly appeared in fantasy literature, with the Narnia and Harry Potter series being perhaps the best known examples. In the myths, they are sometimes wild and uncivilised hunters, and sometimes wise and noble teachers, reflecting their dual human/bestial form; novels have tended more towards the 'wise' version.
While the very earliest Greek depictions of centaurs varied somewhat in which bits were human and which bits horse, the classic look that we're familiar with today was already in place by about the 5th century BC, so it has a long pedigree, and unlike, say, goblins, there's strong agreement on what centaurs are supposed to look like. Both the human and horse parts are often said to be physically attractive for their species, and, while female centaurs appear only rarely in myth, they have been reasonably common in artwork even as far back as the Greek period.
Centaurs have four hit dice, which is more than even the largest of horses; presumably this reflects their combat ability as much as their greater physical bulk than humans. They move at the rate of a medium riding horse, although not that of a typical wild horse, so the human part probably weighs them down. Oddly, given that they don't use armour, they're actually harder to injure than a horse (which itself seems to have a remarkably thick hide); this might reflect a high agility. They are said to be marginally less intelligent than humans, so the intent is likely to emphasise their wild and untamed natures over the myths in which they appear wise.
Generally isolationist, their alignment varies from Neutral to Chaotic Good, something reflected in the relatively small size of their communities, which average 70 individuals, including children. Like many races in 1E, they have strict gender roles, with only the males allowed to even use weapons or (apparently) to leave the secluded forest glens in which they make their homes. Either the females outnumber males by two-to-one in the general population or about half of the latter spend long times away from home. (Which they might, if they live like many real-world herd animals).
Most male centaurs use clubs as weapons, but archers are, as per the legends, reasonably common. In keeping with their 'Chaotic' descriptor, they don't seem very hierarchical, aside from the male dominance thing, and there are typically about four 'leaders' in a band. It may be more accurate to say that these are just the elite warriors, since they are trained to use shields and lances.
As is typical with 2E, the general description of centaurs does not change from 1E, but we do learn more about their society. It's clear now that females really do outnumber males, apparently due to a high death rate among the latter. They practice limited agriculture, and sometimes build shelters in which to live, rather than staying out in the open. Their communities are described as 'tribes', and consist of a small number of co-equal family groups, rather than there being a single defined leader. A very small number of centaurs learn druidic magic, and the race as a whole lives in harmony with nature, avoiding contact with most other races, apart from elves.
Centaurs are as strong as a fully trained warhorse, and somewhat more agile; they apparently have a thick horse-like hide over their entire bodies. Their favoured weapons are now swords, not wooden clubs and lances, which suggests either a rather better knowledge of smithying than one might expect, or a fair degree of trade with the elves. They now are all skilled in archery, rather than relying on specialists.
Community sizes remain about the same as before, but there is a higher degree of sexual egalitarianism now, with females just as combat-trained as their menfolk. Although their society still restricts them to staying at home, centaur women are now apparently the community leaders, and possibly represent most of the spell-casters, which are now more common and varied than before. The explanation as to why they outnumber males is reversed; it's now because the men are always out hunting, not that there's any less of them in the wider population.
Alignment has now switched to 'Neutral Good', implying a more organised society, perhaps reflected in the use of military terminology to describe their hunting parties. The details of centaur language have shifted to make it easier for PCs to converse with them; in 1E they could speak only their own language, in 2E a few in each community could also manage Elven, and now they all speak Elven, and their native language is shared with many other races, such as satyrs and fairies.
Oddly, centaurs are no longer considered to be humanoid in this edition, whereas previously their shape no was no barrier to the use of this descriptor. In practice, this means little beyond the fact that spells like 'Charm Person' will no longer work on them, as they did in 3E, but it's a change nonetheless.
Compared with earlier editions, centaurs have now shifted to a nomadic society. They no longer build much in the way of shelters, and, from the description, are now more likely to be seen in open plains than in forests. Their languages remain unchanged from 3E, so they presumably still deal with the elves; however, while the individual in the picture has a sword, this is now unusual, and lances are now the favoured weapon alongside bows. (Lances require less metallurgical skill than swords, although, since it's hard to see where they'd get the ore from, there's still a decent chance they don't make them themselves. On the other hand, the arrows could have stone tips, for all we know).
Centaurs of the Forgotten Realms are more like the nomads of 5E than the more settled (but clearly non-urbanised) societies implied in the first three editions. This is true even in the 3E sourcebook Races of the Wilds, which is presumably describing the FR version, but differs slightly from the description given in the 3E Monster Manual. Incidentally, the description there says that they regularly wear clothing on their human parts, although the illustrations tend to show that it's only the women who do this.
In Mystara, a few centaurs live alongside human tribes on the fringes of civilisation, but they seem to have had even less impact on most other D&D worlds. In Golarion, they are more commonly True Neutral than 'Good', and not particularly friendly; they apparently have multiple ethnicities, but are most likely to have moderately tanned skin tones. They lack the enlarged ears of the 3E sort, going with a more traditional human look.
Centaurs, being such well-known fantasy creatures, also exist in many other RPGs with a less direct connection to D&D. In Yrth, the primary fantasy setting of GURPS, centaurs are a well-known, if not particularly numerous, race. Their culture does resemble that of the D&D version in most respects, although they are better disposed towards humans. While the majority of Yrthian centaurs are, like those of D&D, less intelligent than humans, a significant minority are born with much higher intelligence, and, like those of Greek myth, work as sages and highly skilled artisans.
In Glorantha, centaurs often have potent shamanic magic, and are powerful guardians of the wild, acting as leaders within the broader beast-man society. On Harn, they are a minor tribal race, found only deep with woodlands and mostly avoiding human contact.
Centaur AnatomyA number of D&D sources state that centaurs have an unusual physiology, although it's never really clear how - it's almost certain that the writers don't know what the word 'physiology' means. However, it's undeniable that they have an odd anatomy from a real-world perspective, and, given that I normally write a biology blog, I'm going to address a few limited points about that here.
In many respects, centaurs have a typical mammalian anatomy, but with six limbs, two chest cavities, and one-and-a-half abdominal cavities. In terms of musculature, nerves, and skeleton, that really isn't much of a problem, although the spine may curve quite sharply at the interface. A number of writers, scientists, and artists have thought about how all the other organs would work, and have arrived at different conclusions as to exactly how many are doubled up, but these are my thoughts.
Centaurs presumably have four lungs, and, since they also have two ribcages and two diaphragms, there's nothing much wrong with that, although the musculature would probably have to be powerful to aeriate the lower pair, and the nostrils and nose might well have to be larger to take in enough air to keep them both inflated. The trachea, of course, would have to continue on past the branches to the upper lungs, and pass through the upper abdomen, but it's only a tube, so that's not really a problem. You will need to synchronise the two diaphragms to breath in and out together, of course, since there's only one nose.
Personally, it seems to me that it would make the most sense to have one really large heart than to have two plugged into the same circulatory system, which as with breathing, you would then need to keep carefully synchronised. (Somebody might point out that octopuses and squid do have three hearts each in the real world, which is true, but this is actually because the four chambers found in the mammalian/avian heart are located in different organs in cephalopods. So, really, it's one heart spread out over three locations, not a triplicate set).
I've seen it argued that, if the single heart were in the horse's chest, it wouldn't be able to pump blood all the way up to the brain, because the head is so much higher up than in a real horse. I think the existence of giraffes rather damages this thesis, although, granted, the heart is going to need to be quite large.
Aside from the need for ridiculously long ureters, there's no obvious reason why centaurs shouldn't have four kidneys, and organs like the thyroid can double up with no issue at all. The digestive system, however, does require some more substantial modification. The centaur mouth is human-like, so they really won't be any good at eating grass (which they do in the Narnia books) and therefore the idea that they might need two parallel digestive systems for different kinds of food doesn't make much sense. Besides, we know they're good hunters, so their diet is likely similar to ours.
It's not inherently unreasonable for food to pass through the human stomach and intestine, and then enter a second stomach and intestine inside the horse part. Rabbits sort of do this, although, in their case, it's by eating their own poo to digest it twice, thus getting more nutrition out of hard-to-digest grass. It is, however, rather unnecessary if you don't eat grass or similarly crappy forage. Real-world horses get round this by having a greatly enlarged caecum, which acts as a fermentation chamber, but if you're going to eat meat, herbs, and cooked vegetables there's really no need.
One could, of course, have two digestive systems that both digest the same sort of food, splitting apart somewhere around the oesophagus, and rejoining down near the rectum. However, it would surely be simpler to have a single, enlarged, stomach in the upper body, that can carry enough food to support the large body, and a long, efficient, intestine (although it wouldn't need to be as long as that of a real horse) in the lower one to extract the nutrition. There's no reason you couldn't have two livers, gallbladders, and pancreases, emptying into the intestine at different points along its length - as with kidneys and lungs, having multiple sets isn't a problem.
The reproductive organs are surely the same as in horses, and, likewise, we only need a single bladder. An interesting question does arise with regard to the navel, which is clearly visible on most centaur illustrations that show that portion of the human anatomy uncovered. It might be that that's the only one, since pictures don't usually show any evidence to the contrary, and this does seem the more likely arrangement, if there's only one heart. On the other hand, if they do have two, fully functional, hearts, then they probably would have two umbilical cords, if only to minimise changes to the layout of the circulatory system.
On a related note, suckling would seem awkward for baby centaurs unless their mother sits or lies down. Otherwise, it would require some degree of contortion no matter which end they choose to take the milk from.
Despite centaurs being a common staple in fantasy RPG bestiaries, they are rarely seen in adventures or other supplements. This is probably because they are a 'good', or at least 'neutral', race that are unlikely adversaries, while their unusual anatomy makes them inconvenient as PCs in traditional games. It would be awkward playing a race that can be defeated by something as simple as a ladder. (Having said this, bariaur, which are ram-centaurs, seem reasonably popular in Planescape. Being smaller than true centaurs must at least help).
I'm inclined, however, to think it's a little disappointing that they aren't used more as NPCs, especially if we play up the 'sage-like' aspects seen in Greek myth, and in novels where lack of dungeoneering ability isn't a drawback. Converting D&D centaurs to other systems often isn't necessary, since they generally already exist with only minor differences.