Not counting orcs, balrogs, and the like. Are there any more diplomatic races found in the history of Middle-earth? Any singular or group of historical significance?
The ones we're obviously aware of:
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In this answer, my aim is to present the races seen in LOTR in the wider context of the Middle-Earth universe.
The first race that was created is the Ainur¹. Think of them as analogs to angels in Christianity, with a strong hint of Greek or Norse gods. They were created before the world by Ilúvatar, the creator god. Ainur comprise two groups, Valar (major Ainur who participated in the creation of the world) and Maiar (lesser Ainur)². In The Lord of the Rings, Ainur don't appear as such, but the three wizards (Saruman, Gandalf and Radagast), the main antagonist Sauron and the Balrog are all Maiar.
Dwarves were created by Aulë⁴, one of the foremost Ainur. Morgoth (also called Melkor, the fallen angel, a Vala, the main villain before he fell and his lieutenant Sauron took his place) created evil races: orcs (or goblins) (evil counterparts of elves)⁵ and trolls (evil counterparts of ents).
In addition, there are many races of more or less sentient beings of unknown origin. The well-known hobbits are supposedly descended from men, but their known history begins well after they became a separate race. Beorn and his family of skin-changers are presumably closely related to humans. Ents, the tree-beings, are a very old race, older than the elves⁶, though very little is known about their origin.
There are also several decidedly non-humanoid creatures that are not dumb animals. Shelob is a giant spider, daughter of Ungoliant of unknown pedigree (the giant spiders in The Hobbit are lesser offspring). Great Eagles are more than birds, but their exact classification was a mystery to Tolkien himself (though they are mentioned in the Silmarillion⁶). Dragons (Smaug, a character in The Hobbit, gets a brief mention) also fit this category; they are evil creatures of Morgoth.
Apart from Ainur, all the major races of Middle-Earth get at least a mention in The Lord of the Rings. There are many more of minor races, subspieces, single individuals (such as the famous Tom Bombadil) and animals of dubious sentience that appear only in one or a few stories. Here are a few pointers to encyclopedic references. Note that you won't find a clear distinction between species, races and peoples, or between sentient and non-sentient in Tolkien's universe.
The Silmarillion begins with two versions of the creation myth: its first two chapters are the Ainulindalë (the creation of the world from the gods' point of view) and the Valaquenta (the gods according to elvish lore); the bulk of the book relates the early history of the world, of elves and men. References (from The Silmarillion):
¹ Ainulindalë: “There was Eru, the One, who in Arda is called Ilúvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made.” Valaquenta: “In the beginning Eru, the One, who in the Elvish tongue is named Ilúvatar, made the Ainur of his thought; and they made a great Music before him.”
² Valaquenta: “The Great among these spirits the Elves name the Valar, the Powers of Arda, and Men have often called them gods.” “With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree. These are the Maiar, the people of the Valar, and their servants and helpers.”
³ Ainulindalë: “Now the Children of Ilúvatar are Elves and Men, the Firstborn and the Followers.” “[The Ainur] perceived that they themselves in the labour of their music had been busy with the preparation of this dwelling [Arda, i.e. the world], and yet knew not that it had any purpose beyond its own beauty. For the Children of Ilúvatar were conceived by him alone; […]”
⁴ ch. 2: “It is told that in their beginning the Dwarves were made by Aulë in the darkness of Middle-earth”
⁵ ch. 3: “thus did Melkor breed the hideous race of the Orcs in envy and mockery of the Elves”
⁶ ch.2: “When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar [animals] and the olvar [plants] (…). before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West. In the mountains the Eagles shall house, and hear the voices of those who call upon us. But in the forests shall walk the Shepherds of the Trees [Ents].”
Since noone else has mentioned them, there's also the Drúedain.
There are also mysterious giants. We know practically nothing of them, they appear in flesh only in the Hobbit, Stone Giants from the Misty Mountains (other references are few rumours said among hobbits during Lotr and legend of giant Tarlang from which came the name for mountain pass in Ered Nimrais, White Mountains in Gondor). Gandalf quote suggest that some of them were ,,more or less decent".
Besides them there are Wargs and werewolves, sentient evil creatures made by Sauron in the First Age: ,,fell beasts inhabited by dreadful spirits that he had imprisoned in their bodies". Werewolves are distinguished from Wargs but they are probably quite similar in appearance, generally they're beings in wolf-shape but much more dangerous than normal predators. Wargs even have form of language (which Gandalf knew by the way) to communicate between themselves, they lived wild in packs just like animals but had intelligence of their own.
And other quite possibly existing species (though it's rather speculative) are vampires, like Thuringwethil, Sauron took a form of a vampire once, so maybe vampire form was taken by evil Maiar or they're different species which form was copied by shapeshifting servants of Morgoth.
Another mention of vampires is during the Battle of Five Armies in ,,The Hobbit" when swarms of huge bats came with Bolg's army:
,,Halt!' cried Gandalf, who appeared suddenly... between the advancing dwarves and the ranks awaiting them. ... 'The Goblins are upon you! ... Behold! the bats are above his army like a sea of locusts.' ..."
,,Soon ... the bat-cloud came, flying lower, over the shoulder of the [Lonely] Mountain, and whirled above them shutting out the light and filling them with dread. ..."
,,Soon actual darkness was coming into a stormy sky; while still the great bats swirled about the heads and ears of elves and men, or fastened vampire-like on the stricken."
Ents and Eagles were created by Yavanna (wife of Aule) with the providence of Eru to protect plants and animals from Man, Elf, and Dwarf.
"All have their worth," said Yavanna [a Valar] and each contributes to the worth of others. But the kelvar can flee or defend themselves, whereas the olvar that grow cannot. And among these I hold trees dear...would that the trees might speak...and punish those that wrong them!" (43).
"Then Manwe [chief among the Valar] sat silent...it seemed to Mawe that the Song [the singing of which created the world] rose once more about him...the Vision was renewed, but it was not now so remote...all was upheld by the hand of Illuvatar [the supreme being]; and the hand entered in, and from it came forth many wonders that had until then been hidden" (44).
"And Manwe said: "O Kementari, Eru [Illuvatar] has spoken saying...When the Children [Elves] awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar and some will dwell therin" (44).
Quoted from the Simarillion
I know this question is old as the hills, but I didn't see petty-dwarves mentioned. We only see Mim and his sons in The Children of Hurin though.
There are Half-trolls and half-orcs. They're not exactly orcs or men. I'm pretty sure someone mentioned this but there was also a mention of demons somewhere, and there are fell beasts. Morgoth's first fortress also held wraiths and phantoms. In the games, there is also a brief view of a Giant Scorpion.
These guys don't show up much, but let's throw 'em in...
There are also the "Nameless Things" living down beneath Moria. This page ( http://tolkiengateway.net/wiki/Nameless_Things ) has a link to a fuller discussion of them, but I haven't seen any canonical source on what they are or who made them.
For now I'll just think of them as some kind of squamous Cthulhoid horrors which are nasty enough that they give both Gandalf and the Balrog the creeps.