While in modern English, the term "bugbear" is really only used to mean a source of irritation or a recurring problem, it did originally refer to a kind of supernatural being. The first half of the word likely has the same origin as the word "bogeyman", and implies a sort of evil spirit. Whether or not the second half was originally meant to indicate that the spirit in question looked bear-like is less apparent, but it was certainly used as such in the Late Middle Ages. By that time, bugbears, like the bogeyman, seem to have been used more as something to frighten children into obedience (such as, say, not wandering off into the forest) than as anything seriously believed in by adults.
The D&D conception of the bugbear is initially an exceptionally large and hairy goblin. Its function in the original editions is as a means of extending the list of five "evil tribal humanoid" races one further step, and thus providing a continuing challenge once characters had reached 3rd level or so. As with those other races, they have become more detailed and varied in use since.
In 1st edition, the five tribal races of evil humanoid form a distinct game mechanical hierarchy, running from kobolds to gnolls. Bugbears are essentially the same idea continued one step beyond the gnolls. In fact, almost the only difference between the two races in terms of game statistics is that the bugbear has one extra hit die, and thus a 5% higher chance of landing a blow in combat and (on average) 5 extra hit points.
They are described as large and hairy goblins, and this certainly fits with the physical appearance. Like goblins, they have broad flat noses and large teeth, although their head appears wider and their forehead flatter. They also have much larger ears, emphasising the "bear-like" look. Apart from the face, they are covered in a thick pelt of hair, which we're told is brown to reddish in colour. The skin is apparently yellowish, and the eyes are pale green.
Unlike the five races below them on the game mechanical ladder, however, bugbears are not described as a tribal race. Instead, the average size of a bugbear community is just over 40 individuals - a clan, perhaps, but certainly no "tribe". As is usual, there are a small number of superior leaders in any group, and the culture is entirely patriarchal, with females only fighting in self-defence. Having said which, the latter are at least competent at doing so, being equivalent to hobgoblins. Mind you, this still implies that they are quite a bit smaller and weaker than the males; the intent may be that, like gorillas in the real world, bugbears are highly sexually dimorphic.
Compared with regular goblins, bugbears have no fear of bright sunlight. They are also marginally less intelligent, have cruder weapons and armour, and are fluent in only four languages, rather than five. Where goblins are "lawful evil", bugbears are chaotic, which would explain their smaller communities. They also have an additional advantage in that they move very quietly, making it easier for them to sneak up on opponents, especially in the dark.
The 2E version lacks the wide dorso-ventrally flattened head of the 1E bugbear and has a smaller nose. While still hairy, they don't seem to have a thick pelt as they did before. Their sight, hearing, and sense of smell are all said to be superior to that of humans. Culturally speaking, there's no real change, however, although we are told that, despite cooperating with goblins, they regularly kill and eat them if they don't find them useful. It wasn't clear in the original version whether their superior armour class was due to a thick hide or simply to better armour, but it's now stated that it's the latter - apparently a mixture of leather and metal pieces that's overall as effective as mail.
Bugbear communities remain about the same size as before, but, oddly, about 60% of the population are now entirely noncombatant. It's not stated whether this includes all the adult females, but it's hard to escape the conclusion that it's at least a high proportion of them. If not, bugbears must either have a huge infant mortality rate or a really short life expectancy after reaching adulthood. Bugbears have also lost the unique language they previously had, now speaking the same tongue as goblins and hobgoblins.
5E bugbears once again change mainly in ways required by the change in the rule system itself. There's also an unusual consistency in the physical appearance, although the individual shown is hairier than they have been since 1E. Nonetheless, there are changes, with bugbears being more agile than before, while returning to the low intelligence of the first two editions. The thick hide has once again gone, compensated for by slightly improved armour, a propensity for using shields, and that higher agility.
There is no mention of them eating goblins (or, indeed, other humanoids) any more, nor any indication that females are weaker or non-combatant. Volos's Guide adds that their communities now rarely exceed 20 individuals, and simple family units, as well as adding that, when they aren't fighting, bugbears are actually pretty lazy.
So, compared with many other humanoid races, bugbears have changed relatively little over the years. They remain a culture of physically powerful, yet surprisingly stealthy, warriors living in clan-sized communities, and unable to cooperate on a larger scale unless somehow forced into it. The changes that have been made have tended to be relatively subtle, and their appearance has also changed little since 2E. By 5E they are only marginally superior to gnolls, but they consistently hold a position one step up from the strongest of the true 'tribal' races in terms of combat prowess and threat.
In most campaign settings, bugbears are either found accompanying smaller humanoids as extra muscle or are found in groups too small to receive much attention in the wider world. Originally described as "giant goblins" they remain one of the three main goblinoid races and are commonly part of their larger culture. Mystara is somewhat unusual in having a moderately sized bugbear "homeland", inhabited by numerous tribal bands united under a single over-chief. Goblins and hobgoblins are also common in this land, holding subservient positions to their bugbear overlords.
In the Forgotten Realms, however, bugbears are more commonly employed as shock troops by the hobgoblin legions and were originally created by them for that purpose. They have the strength, but not the intelligence or motivation, to become leaders over other races, and are at least as often found in small hunting bands. The situation in Kalamar and Eberron is broadly similar, although in the latter a coalition of bugbear tribes has successfully rebelled against their hobgoblin rulers.
The bugbears of Golarion facially resemble the goblins of that world, with a wide mouth and lots of small teeth, rather than large fangs. They are said to have unnaturally large eyes, although the artwork seems consistent in giving them small ones. The "chaotic evil" nature of their society is, however, played up, with them being vicious and sadistic, and even small bands being rarer than they are in other worlds due to their inability to cooperate on a useful scale.
Like the other humanoid races, bugbears are easy enough to translate into other game systems. They are physically stronger than humans on average but are probably only about half-way between the human average and maximum. Where physical bulk is relevant, they are consistently said to be seven feet tall and heavily built. Unusually, no weight is given in 3E, but Volo's Guide gives 250 lbs (110 kg) - a figure that would be really small for an adult male gorilla in the real world and that may, therefore, be an underestimate given their similar physiques.
Bugbearsare agile for their size, with a particular aptitude for stealth and ambush, but are not especially strong-willed. In most versions, they are slightly less intelligent than humans but are not entirely stupid. They also tend to lack any natural armour and prefer to use spiked clubs as weapons. The primary advantage of bugbears over other humanoids, however, is their superior hit dice, something that doesn't often have a direct correlate in other systems. This is probably best reflected in a high combat skill, and perhaps some skill at dodging or parrying blows to reflect their high agility.