Well, let's see, We have:
Boldogs: Lesser Maiar who have taken the shape of large Orcs.
Orcs: general term for any orc or goblin; also uruks (Black Speech) or yrch (Sindarin; orch is the singular form).
Hobgoblins: According to Tolkien, a smaller variety of goblin inhabiting the Grey Mountains.*
Snaga: Literally 'slave'. Perhaps not a distinct breed, but smaller orcs that get bossed around a lot.
Half-orcs and goblin-men: Hybrids bred by Saruman.
Uruk-hai: Sauron's improved orcs; man-sized and not hindered by the sun.
* Thanks to eldariontkd for reminding me of hobgoblins from The Hobbit.
Hobgoblins were also mentioned in the Hobbit, but that may be just another term for goblin. If goblins are the smaller, weaker orcs, then hobgoblins may be the Uruk-hai.
Thanks for the reminder. Not sure if hobgoblins are supposed to be larger or smaller than other goblins, The etymology suggests 'smaller', and Tolkien in a letter to a Roger Lancelyn Green (http://tolkiengateway.net/wiki/Hobgoblin) indicated that that what was intended.
Tolkien in The Hobbit also mentions ogres, but they seem to be more closely related to giants (and maybe trolls).
I'm used to the D&D convention where hobgoblins are larger and stronger than goblins and orcs (in the earlier editions, at least). The old-school D&D equivalents were:
Goblins = Goblins of the Misty Mountains
Orcs = Uruk-hai of Mordor
Hobgoblins = Saruman's Half-orcs
D&D orcs and goblins don't seem to be closely related though. And D&D also has the even smaller kobolds.
So, if D&D hobgoblins are the equivalent of Half-orcs, what are Tolkien's Half-orcs in D&D? And what about D&D bugbears?
Hobgoblins were around way before the half-orc appeared in D&D, so the separation came around at that point. Kobolds and bugbears come from folklore, but they appear quite different from the D&D depictions. Some stories depicted kobolds as helpful spirits, warning miners of possible cave-ins.
True, but orcs weren't described as having snouts until the Monster Manual, which came out in 1977.
Here's a painting done by the Brothers Hildebrandt.
It may have been the inspiration for orcs in early D&D.
Games Workshop made all orcs and goblins green.
As for modern depictions, well....
It might be possible that the painting by the Bros. Hildebrandt inspired the appearance of D&D orcs; I haven't found a date, though, for when the painting was produced. The game goes back to 1974.