The term "hobgoblin" is a part of traditional British folklore, referring to a particular sort of goblin that's usually seen as less malevolent than the normal sort - albeit capricious, and often dangerous pranksters. Puck, in A Midsummer Night's Dream, is a hobgoblin of this sort, showing that Shakespeare, at least, thought of these beings as mischievous, but not actively evil. The original meaning of "hob" is unclear, although there's nothing to suggest that it originally meant "larger".
Tolkien is the first to use "hobgoblin" in that sense, using it, briefly, to refer to larger orcs in The Hobbit, in distinction to the regular "goblins". Gygax presumably borrowed this for D&D, and seems also to have been influenced by the uruk-hai of The Lord of the Rings, which share a number of traits with 1E hobgoblins.
Although originally conceived of as large goblins, the first edition of the Monster Manual gives no indication, beyond the obvious matter of the name, that goblins and hobgoblins are particularly related. Being slightly larger than humans, the latter are over half again the height of goblins, and don't seem to have much in the way of physical resemblance to them. They are said to be hairy with brown to grey skin over most of their bodies, but with bright red (and apparently hairless) faces. Large males have blue noses.
The function of hobgoblins in game at this point appears to be simply be a slightly tougher version of the orc. The second highest in the clearly-defined hierarchy of evil tribal humanoids, compared with orcs they have one extra hit point, a 5% higher chance of landing a blow, and one extra point of armour - largely the exact opposite of goblins. As is typical of these races in 1E, a proportion of the males are significantly tougher, the females are apparently non-combatant, and there is no indication of magic use among them.
Perhaps the most notable difference, however, is that, unlike the other four evil tribal races, they are said to have fully human intelligence. They are all fluent in three different languages, which is actually less than the supposedly stupider orcs and goblins; a few can also speak the Common Tongue. More significantly, they are specifically stated to polish their weapons, and the illustration shows a being wearing something like samurai armour (apparently equivalent to mail) over decorative clothing. It's probably this, more than anything else, that led to the assumption that they are relatively sophisticated culturally, which was followed up more in later editions.
Although they are typically subterranean, they lack the fear of sunlight that the weaker three races possess. Hobgoblin communities average around 600 individuals, but this is largely due to children outnumbering the adults - implying a massive infant mortality rate. Uniquely, females outnumber the males, forming 60% of the adult population; this may imply that they are less likely to get themselves killed in battle.
Powerful evil individuals are said to use them to enforce discipline among orcish and goblin troops, but there's otherwise no specific association with the latter. They do, however, commonly associate with carnivorous apes, and can even communicate with them on a rudimentary level. A green-skinned, water-breathing, variant of the hobgoblin, the "koalinth", is also mentioned, but not much elaborated on.
There are no significant changes between the first and second edition versions of hobgoblins, with much of the text from 1E being copied across almost verbatim. In terms of appearance, the 2E hobgoblins do have much larger ears than their 1E counterparts, but that's about it. The example in the illustration is now seemingly more influenced by Mongols than Samurai, and we're specifically told that their society is organised and militaristic. (In this light, it's interesting to note that, since females outnumber males, most of them aren't warriors. Presumably, they provide support to their menfolk, but unless they're significantly weaker, it seems less likely that they are mere chattel as female orcs and goblins are).
The hobgoblins of 3E are similar in general appearance to those of 2E; the one in the picture has a beard, but it's not unreasonable to suppose that those in the first two editions happen to have shaved rather than being naturally beardless. They are now described as having the same skin tone all over, which is more orange than red according to the text, and more of a tan shade in the illustration. The armour they wear looks fairly sophisticated, although it's supposedly no better than studded leather.
The description of their culture greatly plays up their militaristic mindset, pointing out that they are highly disciplined and capable of employing sophisticated tactics. Because of this, unlike orcs and goblins, they retain the "lawful evil" alignment that the other two races have now lost. The society turns out to still be patriarchal, with only males allowed to form warbands and take leadership positions. However, the females are now also warriors, albeit in a subordinate position focused primarily on defence of the tribal home and their children. (The latter are much less numerous than before, although there are still enough to suggest high birth and infant mortality rates).
The former association with carnivorous apes has now gone, with their role taken by ogres, and dire wolves being common in the larger tribes. The text also states that they are "often" leaders among goblin and orcish tribes, apparently even they aren't working for Evil Overlords™. Their much closer association with goblins, in particular, is indicated by the fact that they no longer have their own language, using the goblin tongue instead.
The original step-wise progression of the evil tribal races has gone by this edition, with hobgoblins now being marginally weaker than orcs in most respects. This is, however, compensated for by an even stronger emphasis on their martial culture, describing them as a particularly effective fighting machine that's no longer truly "tribal". Their legions regularly employ longbows and large siege engines, and the general implication is that hobgoblins are now as civilised as anyone else, albeit with a civilisation based entirely on war.
The physical description is basically the same as in 3E, although the picture now matches the text in terms of skin colour - the warlord in the picture also has a much narrower chin than his counterparts in previous editions, but that could just be him. They have, however, shrunk, now being the same height and weight as a typical human. Hobgoblin armour is now also the equal of anything humans would typically use, typically being mail (rather than reinforced leather), and with leaders getting full suits of plate in a distinctly samurai style reminiscent of that in 1E.
In other respects, ruling over goblins now seems the default for hobgoblins, and, for the first time, bugbears are also regularly included in their legions. In a nod to older editions, carnivorous apes make a comeback, although they appear to be unusual additions to the troops, with giant riding wolves being more common, along with more standard beasts of burden, and ravens for communication. It's implied, if only by omission, that the sexes are now of equal status.
With hobgoblins originally being larger, more intelligent, daytime-active, versions of orcs, it's clear that the original intention was something similar to Tolkien's uruk-hai. From here, perhaps in part due to the initial art choices, they have evolved into a steadily more sophisticated - if no less hostile - culture. While every published version in the standard rulebooks, and in the default setting of Forgotten Realms, has stated that different hobgoblin tribes are mutually hostile to one another, a number of other settings have extrapolated their orderly, militaristic lifestyles to the creation of organised, long-lasting, nations.
While I've seen it more than once in home-brew settings, the first published setting to do this was probably Mystara. This shows a fairly tribal culture, based loosely on the Mongols, but enclosed in a relatively narrow area, where they are at war with the local orcs. Since the other tribal races have similar nations, there's perhaps not much particularly distinctive about that of the hobgoblins in this instance.
Kalamar has no less than two hobgoblin nations, speaking related, but mutually unintelligible languages, and with only small minorities of goblins and other races. Each nation supports one major, and one minor, city, which suggests a high degree of organisation. As in the Forgotten Realms setting, hobgoblin wizards also exist, being used solely as military support. Unlike that setting, however, we're specifically told (rather than it just being implied) that females and males have equal standing - although all the main leaders to seem to be the latter. Interestingly, hobgoblins in this setting can crossbreed with humans, although it seems to be rare.
In Pathfinder, hobgoblins are grey-skinned, distancing them from the standard D&D appearance. They are stated to be hairless, although the actual illustrations show that most females, in particular, have a full head of hair. Or possibly they just like wearing wigs. They have forged at least one, and possibly two, nations in the default setting of Golarion, in which it is stated that the goblins are the original race, and hobgoblins a magically created offshoot. They themselves despise magic, but are intelligent enough to be good at science - albeit, given their interests, that mainly means the sort of chemistry that makes things go 'boom'.
Imitating D&D-style hobgoblins in other systems is largely a matter of defining an effective soldier. They are, on average, stronger and tougher than the typical non-combatant human, and in most other respects are largely similar. Being well-disciplined, the fact that they are evil doesn't present the sort of social disadvantages (e.g. uncontrolled bloodlust) that it does for other similar races. In terms of statistics, it's probably more a case that they're all at least moderately efficient soldiers and never have more empathic traits, than that they're really all that different from typical human warriors.