Given that crawling around in subterranean labyrinths is part of the point of the original D&D, it's unsurprising that the Minotaur would be included in the game. Here, of course, it becomes a race of beings and thereby loses its capital letter. (As an aside, both "mine-otaur" and "minn-otaur" are legitimate pronunciations in UK English, although the latter seems to be preferred in the US. In Ancient Greek, it was apparently "meen-otaur", so, hey...).
Minotaurs in 1E are significantly larger than humans. By how much isn't clear, although they can't head-butt anything shorter than six feet, which may imply a height not much more than that. They have, of course, the head of a bull, but with a full set sharp, carnivore-like teeth and lower canines that form tusks. Unlike most classical depictions, they have a thick pelt of fur over the torso and upper arms, although we don't know the form of the legs, or whether they have a tail.
Minotaurs are as strong as ogres, and even more skilled in combat, although it is slightly easier to land a blow on them, implying a thinner, if still substantial, hide. Unlike the lumbering ogres, they move with human-like speed, and they seem to be even more aggressively violent. They are described as "unintelligent but cunning", a not unusual combination in D&D, and they are also noted to have acute senses of smell and (probably) hearing.
Like the version from classical myth, they have a craving for human flesh, although, absent cooperative Cretans, one assumes that that can't be in terribly high supply in the underground labyrinths that they inhabit. However, while they're generally antisocial, they aren't entirely solitary, and can often be found in groups of four or five. That they also have their own language implies some degree of culture and traditions among them, although presumably at quite a basic level.
The physical look changes little in 2E, although they seem to have lost the tusks. We can see that their feet are at least generally humanoid, although it's possible that they don't have human-like toes. The only significant change to their abilities is that they now have an unusual affinity for navigating mazes, and are immune to magic that might hamper this ability. This, of course, is the exact opposite of the original myth, where the whole point of the labyrinth was to stop the Minotaur getting out. They also have an ability to see in pitch blackness, which does rather make sense given where they live.
As for society, a simple "clan"-like culture is described, with the oldest and most physically powerful minotaur leading the others. The labyrinths they live in are apparently built for them by evil humans - presumably to use them to guard something else, although the details are vague. Judging from the picture, the evil humans in question also provide well-crafted clothing and weapons for the minotaurs, since they clearly can't make such things for themselves.
There is a radical change in the minotaur's look in this edition. The head is now ape-like, rather than bull-like, aside from a pair of huge sweeping horns. The legs end in large cloven hooves, the tail is relatively large and the creature is much shaggier than before. Finer detail in the base attributes in this edition means that we can now see that minotaurs are marginally weaker than ogres (although still around human maximum), but slightly more agile and not quite as stupid. They retain broadly the same special abilities, tweaked for the new rule set. It also turns out that their hide is the same thickness as that of ogres, but that the latter tend to wear thick leathers as armour.
They do, however, live in smaller groups than before, with no sign of the eight-member "clans" of previous editions, and often being solitary. They now speak the same language as giants, although quite why this should be when they seem to have no obvious connection to the giant races is a bit of a mystery.
With 5E, we return to the idea of a minotaur actually looking part-man part-bull. The key differences from the 2E version are the cloven hooves and the fact that they only have three fingers and a thumb on each hand. Compared with the 3E sort, they have lost a point of intelligence, but so have ogres, so the two remain the same in relation to one another. Unlike ogres, minotaurs retain their tough hide in this edition.
The steady decline of the size of minotaur "communities" continues in this edition, with them now being primarily solitary, perhaps better echoing their mythic roots. Indeed, we're told that they are impossible to keep even as imprisoned slaves, which would suggest that the 2E association with evil overlords just won't work. Their native language has changed again, this time to Abyssal, which makes sense given the origin story we're provided with.
Undeniably the biggest question about minotaurs, biologically speaking, is where new ones come from. Early editions of D&D are quite clear that all minotaurs are male, something that obviously doesn't make sense given real-world biological norms. The core rulebooks ignore the question in both 1E and 3E, although that's not always true in individual game settings for those editions. But, if we want to answer it, three options present themselves.
One is that minotaurs are the result of some kind of curse. This is the case in the original legend, where the Minotaur is the result of a curse placed on King Minos (by punishing his wife, but that's the Greek gods for you). It's also the standard origin in 5E, where one can become a minotaur through demon-worship and it's at least the historical origin of the race in a number of specific game worlds.
2E officially went for a different, and far less palatable option, whereby minotaurs breed with human women and all the offspring are male. Since it's hard to imagine that many sane human women would be happy about this, it's not hard to see why this suggestion was dropped like a hot potato in later editions...
The third option, of course, is that female minotaurs do, in fact, exist. That was unnecessary in Greek myth, since there was only ever the one, but it's obviously the easiest solution if you want to have a race of the beings. In fact, it's the usual option taken in D&D game worlds, even if pictures of them are rare outside of mildly dodgy "furry" fan art - although not, to be fair, non-existent. Both Mystara and Golarion explicitly have female minotaurs and even the 5E Monster Manual states that minotaurs can breed true if they ever encounter one another, which clearly implies the same thing (and the text is gender-neutral, without a mention of "bulls").
Perhaps the only oddity of D&D minotaurs, compared with those in other fantasy RPGs, is their inherent ability to escape from any maze in which they might be trapped - and even that is implied in, for example, GURPS Banestorm. In that system/setting, they have the Absolute Direction advantage and that, or some sort of special boost to navigation skills, would be the way to go in many others, too. Otherwise, D&D minotaurs have approximately maximum human strength, around the minimum human intelligence, unusually acute senses, considerable combat skill, the ability to gore or charge with their horns and at least an impulsive recklessness, if not outright berserk rage, in battle.