When we first meet the Balrog in Moria during The Fellowship of the Ring, Gandalf explains that the creature is "a demon of the ancient world." The way that he phrased that gave me the impression that Balrogs are a type of demon, which makes me wonder if any sorts of demons besides Balrogs exist in The Lord of the Rings canon. If so, what are they?
Technically the main type of "demons" (which are not the same thing as devils) in LOTR are more or less evil individual Ainur, spirits created before the world was made. The Ainur are pure spirits, but they can create physical bodies for themselves as easily as Elves and Men can put on clothing.
Many of those spirits entered the world when it as made. Their leaders are the great and godlike Valar, while the lesser Ainur that came to Middle Earth with the Valar are called Maiar, and Ainur remaining outside the world may have other unknown categories.
The first dark lord was Melkor, mightiest of the Ainur, later known as Morgoth, and his followers included many Maiar he corrupted as well as other Ainur he lead into the world. Sauron was a Maia serving the Vala Aulë until Morgoth corrupted Sauron.
It is possible that some "spirits" in LOTR are not Maiar, or Valar, or any other type of Ainu, but totally different types of spirits. Some beings are very hard to categorize.
We must, I think, take "demon" in the Catholic sense here. Tolkien viewed his alternate world as consistent with Catholic theology. Now, there have been some changes since Tolkien's time, but according to the Catholic Encyclopedia, which:
[...] bears the imprimatur of the Most Reverend Archbishop under whose jurisdiction it is published.
The Encyclopedia says, of the word "demon,"
In Scripture and in Catholic theology this word has come to mean much the same as devil and denotes one of the evil spirits or fallen angels.
So in Tolkien's legendarium, a demon must refer to an immortal spirit corrupted into evil.
Thus a demon would be any Ainu or other spirit that had followed the corruption of Morgoth, and of course Morgoth himself. There are several classes of evil spirits that followed Morgoth (besides of course the Balrogs).
Morgoth himself, as well as Sauron, did not seem to belong to any larger class of demon. They were an evil Vala and Maia, respectively, and were the two spirits to set themselves up as Dark Lords, ruling over other spirits and mortal races. Ungoliant, a monstrous spider spirit, probably also qualifies. Technically, even Saruman might have been considered a demon by the end of the LOTR trilogy.
Thuringwethil and other vampires were probably Maiar who took on the form of enormous bats. Certainly both Sauron and Thuringwethil assumed this form from time to time.
The werewolves, or at least some of the greater ones, may also have been Maiar. The LOTR wiki suggests this:
Werewolves were servants of Morgoth, bred from Wolves and inhabited by dreadful spirits (fallen lesser Maiar or the fëar [souls] of Orcs) imprisoned in these wolfish forms by Sauron.
Certainly some of more powerful werewolves may have been Maiar (and thus demons)
Then Sauron sent Draugluin, a dread beast, old in evil lord and sire of the werewolves of Angband.
In The Silmarillion, "demon" always refers to servants of Morgoth, and Balrogs are the only demons mentioned by name.
The Silmarillion's Index of Names and appendix give the etymology of "Balrog" as "Demon of Might," from bal- (power, might) + rog (demon). The Quenya word for "Balrog," Valarauko, can be broken down in the same way, as vala- (power) + rauko (demon). So the etymology implies that other types of demon are possible: a Balrog is a type of demon.
This passage from the Valaquenta in the Silmarillion shows that the Balrogs were Maiar who sided with Morgoth, but they were not the only Maiar to do so:
For of the Maiar many were drawn to his [Morgoth's] splendour ... Dreadful among these spirits were the Valaraukar, the scourges of fire that in Middle-earth were called the Balrogs, demons of terror.
The Balrogs were among the Maiar that were drawn to Morgoth, but there were others.
Also from The Silmarillion, in the second paragraph of chapter three (emphasis mine):
And in Utumno he gathered his demons about him, those spirits who first adhered to him in the days of his splendour, and became most like him in his corruption: their hearts were of fire, but they were cloaked in darkness, and terror went before them; they had whips of flame. Balrogs they were named in Middle-earth in later days. And in that dark time Melkor bred many other monsters of divers shapes and kinds that long troubled the world.
The Balrogs were among the first spirits to join Morgoth, and the most corrupted, but the text implies that other spirits also followed Morgoth. The way I read the passage, it appears that the demons are spirits. The other creatures that Morgoth bred were simply called monsters. If all demons are spirits, and all demons followed Morgoth (at least until his banishment), then a lesser Maia who followed Morgoth would be a demon, rog, but not a Demon of Might, Balrog.
After the valar created the sun and moon in The Silmarillion, Morgoth sends "spirits of shadow" to attack Tilion, the moon-man. Note it mentions they were spirits of shadow, but not fire. So maybe these were lesser balrog-like Umaiar (demons of Morgoth) and cloaked themselves entirely in shadow. In the same story it is also stated Arien, the sun Maia "was a fire spirit from the beginning" perhaps implying the Maiar that Melkor corrupted were offered the chance to be fire-spirits themselves. But it's just as possible that they were just fire spirits who aligned themselves with Melkor. And Sauron is not one of them because he was a forging-spirit, a craftsman of Aulë, not a spirit of flames. But this all just my own opinion.