Centaur
A race of monstrous humanoids, originally from Greek mythology, best known for being a literal mix of man and horse - they have a human's upper torso (that is, the body from the top of the hips up to the top of the head) replacing the head. They appear in various fantasy settings throughout the world now.
Who were they?
The Centaurs, as mentioned above, were a race of mythical creatures from Greek myth, noted for being horny bastards and massive drunks. They were one of the less important monsters of their myths, with only a handful of references. Even so they did get a constellation.
In their creation myth, the centaurs were descendents from a mythical culture called the Lapiths, via their douchebag king Ixion. In short, Ixion was an asshole who murdered his father-in-law by burning the dude alive, a huge no-no for the Greeks. Somehow, Zeus decided to be a super-chill guy, and not only forgave Ixion, but actually brought the asshole to Olympus to have dinner with the gods. Did Ixion realise just how lucky he was? Nope; asshole started eying Zeus's wife, Hera. Zeus got suspicious, so he made a double for Hera out of clouds, called Nephele, and had her take Hera's place serving Ixion some dinner. Ixion promptly attacked and raped Nephele, for which Zeus smote his ass with a lightning bolt, then tied his spirit to a burning wheel and sent it spinning endlessly through the "sky" of Tartarus. Nephele got pregnant and she gave birth to the Centaurs.
Ironically, one of the Centaur myths describes their interactions with the Lapiths. The Lapiths invited the Centaurs to one of their weddings, and then things went haywire very, very fast. Though the Centaurs had been benevolent beforehand, the copious amounts of wine being served got them all drunk, which meant they turned on a plate during the wedding, trying to rape the women and kill the men of the Lapiths. This ended up as one huge brawl, ending with a Lapith victory. This was the Greek way of saying "Hey, nature? You suck.".
Chiron, a great healer and teacher of heroes in Greek myth, was a rare case of a friendly centaur.
A centaur was also responsible for slaying Hercules (or, to give him his original Greek name, Heracles). He tried to kidnap Hercules' wife Megara to rape her at his leisure, and Hercules shot him with one of his poisoned arrows for it. As he was dying, he told Hercules' wife to secretly soak a cloak in his blood, promising that if Hercules' eye started to wander, all she'd have to do is make him wear it and he'd fall back in love with her. She believed him, but the gory cape was actually poisonous and drove Hercules so mad with pain. In one version it caused him to kill her and their children then awaken and see what he'd done causing him go on another adventure to atone for it, in another he simply burned himself alive on a funeral pyre to find relief.
If you want to know about the real world origin of the Centaur, well consider where Greece is. Go too much farther north and your hit the Ukrainian and Russian Steppe where the horse was first domesticated and where nomadic tribesmen practically learned to ride before they could walk. It's believed that the early Greeks saw either early horsemen or the first horsemen at a distance and could not understand that they were looking at a human ON an animal, (a distinction that the Spanish report the Aztec had trouble grasping) and instead saw a horse with the top half of a man. Hence the origin of the Centaurs.
Biology
Centaurs are a fascinating example of a complete can of worms when looked at from a scientific standpoint, mainly because the way the Centaur is structured consists of two torsos (removing the horse torso would make it a Satyr, removing the human torso would make it a horse-themed Sphinx).
In theory, a centaur has two different sets of ribcages. Depending on how the setting has the internal anatomy work, a horse could have two pairs of lungs, two hearts, an extra stomach and possible extra intestine leading to the horse stomachs, two livers, two sets of kidneys, and most anatomy exclusive to both dedicated carnivores and to dedicated herbivores. If this version of Centaur actually existed, it would possibly be the most unspecialized and energy-efficient diet in the setting. More often however, the human half is explained to have large lungs and a large heart while the rest of the anatomy is contained within the horse half. This presents the problem of having a ridiculously long esophagus which has an elevated risk of choking while the Heimlich Maneuver would have greatly reduced ability to save the Centaur's life, although as long as food manages to make it past the human half the rest of the esophagus would not require more than a drink of water to alleviate discomfort.
They do not need to wipe their own asses, because their horse anus wipes itself through layers of sphincters. Any additional cleaning for social reasons however would require a sponge or towel on a stick. Likewise, putting garments on the horse half for reasons of modesty would be extremely difficult.
Human babies are born completely helpless and unable to do much of anything by themselves other than emit noise. Horse babies immediately run once the birth awl is cleaned off by the mother. In theory, this means centaur babies run around and play while the human half flops around helplessly. More likely is that centaur babies would have both halves as developed as that of a regular horse since the horse womb's larger opening gives more time for the head to grow.
On /tg/
Centaurs invite ceaseless argument over whether they count as monstergirls or as furry. On the one hand, they do have human-like heads and not muzzles (though Japanese hentai seems to like portraying them with horse ears instead of human ears, and this has become more common in western art too). On the other hand, they have horse-junk at their business end (except, again, for some Japanese hentai, where they have human pussies at the front of their horse bodies instead or in addition to mare-cunts at their ass-end), which makes them in many ways more like bestiality than regular furries.
Not helping things is that both sides of the divide have embraced them. In fact, centaurs are so popular that the idea of replacing the horse with other critters that sound cool or sexy; lions, wolves, tigers, spiders... heck, lamias can be described as "centaurs that are half-snake instead of half-horse". The idea is so popular that a whole new word (arguably descended from the furries), "tauric", has popped up as a shorthand for it. /d/ alternatively loves them (because they're an excuse for pictures of huge buff guys with huge horse cocks, monstergirls with huge-ass tits, or monstergirls with huge-ass tits AND huge horse cocks) and loathes them (because of the aforementioned furry/bestiality connotations).
Surprisingly, despite the fact that centaurs are iconic creatures from western "Classical Mythology", the vast majority of centaur MG depictions have come from Japan. This may be because, unlike westerners, Japanese smutfans don't really make so much of a distinction between furry and monstergirl, and so they don't get bothered by the whole "you're fucking a horse's hiney" thing since they pass the Harkness Test.
Perhaps because of the strong association between the cavalry and the samurai caste, or because Japanese dig soft, sweet chicks to an almost disturbingly degree, almost all known Japanese centaur MGs have been tsunderes or yamato nadeshikos - that is, either sweet-girls who act like bitches because they don't wanna admit their feelings, or very sweet and demure girls with a steely core. A more mythology-accurate centaur MG would be a total ladette - a she-jock who loves to fight, hunt, drink and fuck like there's no tomorrow and aggressively chases after good-looking guys. Sort of a bro with the body of a super-model who's fine with letting you grope her cunt & titties.
Warhammer Fantasy
The closest example fantasy has for Centaur are Centigors. A type of Beastmen whose upper part is that of a Gor (humanoid goat), and the standard lower horse part. Centigors are the Beastmens primary cavalry unit, they are good as shock troops and charging into the fray at lighting speed. Now if only their stats match their lore.
If thats not enough then things get bigger. With Dragon Ogres, because regular centaur isn't awesome enough. Dragon Ogres have the upper part of an Ogre and their lower parts are four dragon like legs. These Ogres made a pact with the Chaos Gods that they would serve them, in exchange for immortality (at the cost of having no female Dragon Ogres to fuck with, or possibly having females but all being sterile; it's hard to say from the fluff). They are one of the oldest creatures in the Old World, and are supercharged when struck by lightning. The oldest and largest Dragon Ogres are Shaggoths.
In between those two are the Bull Centaur, which are half Chaos Dwarf and half bull, which are the result of worshiping of Hashut, the bull Chaos God. They are big, mean, muscular, have horns on their heads and curly beards like the rest of the Chaos Dwarfs.
Warhammer 40K
Though the Centaur in and of itself haven't been discovered as of yet in the dark millennium (it is perfectly possible that they exist somewhere in the galaxy, mind you - there is weirder shit out there), the Interex had a heavy infantry platform which had the shape and appearance as the Centaurs of myth. The human soldier was fit into the front of a mechanical, horselike body, (where the head would've been on a real horse) so only the upper body was visible - They even went so far as giving the soldiers laser-crossbows.
Though most would argue that this was freaking awesome, they were exterminated by the Imperium, and as far as we know, the Mechanicum haven't kept anything from the Interex... But who knows, maybe the next Imperial Guard Supplement will include cyborg horse-human hybrids (of course you can make these anyway and just proxy them as rough riders on the table top). You can't know with GW these days.
Centaur body is one of the common Chaos mutations, though, given the nature of Chaos you may end up with a snake tail, giant bloodhound body, or a fucking spider torso instead of your legs, and that's if you get lucky - some poor Nurglite sod may get snail or maggot body instead of his legs, or even a twisted mess of tentacles that have the mind of their own and immediately start raping all asses in proximity, including their own.
Additionally, an Imperial Guard short story (Children of the Emperor) from the Into the Maelstrom anthology features a guardsman stranded on a lost, feral world with super-high gravity, populated by abhumans whose ancestors gave themselves centaur bodies and super-enhanced muscles to cope with high gravity and abundance of deadly predators. He gets involved with a struggle between the Emperor-worshipping locals and a Khornate heresy, but the best he can do is to report to the Imperium when he escapes that the planet is feral, too dangerous & resource poor to colonize, and houses only some unimportant alien species, thus preventing the loyalists from being slaughtered as mutants by more Puritanical Imperials.
Also, the Zoats, a now-abandoned race of reptilian taurs that were originally envoys and propagandists for the Tyranids, before they fully embraced the Horde of Alien Locusts thing.
Dungeons & Dragons
Centaurs and other "tauric" creatures have appeared throughout the editions and settings of Dungeons & Dragons, from the half-lion wemics of Forgotten Realms to the Driders that have become so iconically linked with the Drow.
Centaur are rarely used as Player Character races due to the difficulties of a Centaur performing non-running feats of agility, but those players who do use them find that they are both quite powerful and also have the perk of not having to buy a mount.
Warcraft
Warcraft features many types of Centaur as well as centaur-like creatures. The earliest Centaur are descendants to the demigod Cenarius (half-Night elf half Stag), his sons are called Keepers of the grove, they have the upper part of a night elf with antlers on their heads, one of their hands is wood like, and they have the lower parts of a stag. The female versions of them are Dryads who are female night elf and half doe.
They also have regular Centaurs who are a bastard race born from a Keeper of the grove and (the rather ugly) spirit of Earth, they thank their father for creating them by killing him. They have the traditional Centaur look, with a nomadic Mongol-like culture mostly built on harassing the cow-like Tauren.
Life With Monstergirls
Centaur have a long and noble history as knights and paladins, and most of their race still carry themselves with dignity. Female Centaur often adopt knightly demeanor and customs. Their females are only fertile during certain periods of the year, and must use this time to produce offspring. Their sex drive is greatly reduced, although not non-existent, outside of estrus. Centaur long had a history where the sexually selected traits for males was a general raw strength and bravado. This unfortunately resulted in most modern Centaur men being unattractive fratboys. Recently, their culture is shifting towards effeminate and polite men being attractive and this means that Centaur women are beginning to more and more desire human men. The product of a human/Centaur union is a full-blooded Centaur, but the practice is still highly taboo. Instead Centaurs have been relying on human men as "teasers", a role that exists in real horses. The human man more or less arouses a female who is in estrus until she can bear to reproduce with her husband.
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Centaur sexual habits in Daily Life with Monstergirl
Monstergirls
In the Monster Girl Encyclopedia, centaurs are a race of tsundere nomadic archers. They do still love booze, though, and have a tendency to lose control and become horny drunks who'll rape men... of course, being tsunderes, they'll bitch at the men to "take responsibility" after they sober up. For many years, constant arguments frequently sprung up about where the genitals are. The author decided to split the difference by decreeing they have two sets of genitals, one on each end. Fans are split as to whether this succeeded in solving the problem or is the stupidest thing in the history of ever.