We are told many times that the Dark Lord(s) cannot create - they can only twist and imitate. The orcs were made in mockery of the Children of Iluvatar; the trolls are imitations of the Ents.
So... how far does this connection go? Let's assume that Morgoth - or rather, Melkor, back during the ages of the Lamps and the Stars - is utterly incapable of creating intelligence, that he is only able to twist things that already have it, and furthermore that he chooses to imitate and 'improve' what he sees around him. This would be entirely in character for him - attempting to build everything in the world from as near to scratch as possible, to prove that he is the true creator of it all.
Melkor's armies consist, at various times, of the following:
Wargs (and werewolves, which I believe are the same thing)
Humanoid Maiarin lieutenants
That's ten 'species', loosely speaking, though there may be some overlap between them. Among the non-Melkor-created (though not necessarily good) species, the following are intelligent:
Iluvatar (if we can even call Him a species)
The humanoid Maiar
Nature spirits (like Tom Bombadil, and quite possibly Ungoliant and Huan as well)
That makes nine. Is it possible, then, to line these creatures up, and identify what Melkor was imitating with each?
Melkor. Given his character, and the ideas he spread, it's fairly clear that he saw himself in the role of the One, Iluvatar. Blasphemous in the extreme, but that's Melkor for you.
Lieutenants of Melkor, such as Sauron. Given who the Dark Lord has set himself up in imitation of, Sauron can only be seen as a fake Vala. I bet he loved that.
Vampires, the iron-clawed possible bloodsuckers who functioned as messengers. Like many of Melkor's servants, they are probably Maiar themselves - and I think they function as a reflection of Melkor's views of the good Maiar. The nasty, fragile and irritating creatures who swarm about their masters, sneaking around carrying messages - yes, I can see that as Melkor's view of the likes of Eonwe.
Orcs. We already know that these are probably twisted elves and men, and therefore can be said to be a mockery of the Children of Iluvatar as a whole.
Trolls. Again, Treebeard tells us they're a parody of Ents - though whether they come from torturing Ents (or Entwives, I've seen that theory) is unclear.
Wargs. We know for a fact that both Sauron and Carcharoth took their forms in imitation of Huan. The Wargs are also one of the most 'natural' of Melkor's servants - you can't really see vampires raising their young, but wargs, certainly. I would class these as twisted and debased nature spirits of some kind - what kind, precisely, I don't know, because Tolkien said so little about them. It may even be the case that wargs, like giant spiders, are a natural intelligent creature, and that werewolves are the 'mockery' of them.
Cold- and fire-drakes. I class these together because they feel very much like a progression of technology: at some point, Melkor learnt how to add fire to their abilities. These gold-loving, cave-dwelling creatures are a clear mockery of dwarves - where Aule the Smith crafted imitation Children, Melkor the Lord of All crafted something so much better. (There is actually a 'theory' somewhere - I can't find it any more - that dragons are dwarves - an entire colony of them, all mashed together. Ouch)
If you like, you could even imagine that the Cold-drakes are in imitation of the wandering Petty-dwarves, while the Fire-drakes imitate their forge-using cousins.
Winged dragons. Given what Melkor saw every time he looked out of his window, Ancalagon and his brood are obviously an answer to the Eagles. Unlike all the other 'fakes', though, they still retain the nature of their wingless predecessors, hence Smaug.
Balrogs. Finally balrogs. But there is nothing left to compare them to - I've exhausted the list. Except... before the Eldar began their great March west, three of their leaders were taken to Aman - Ingwe, Finwe, and Elwe. When they returned, to the eyes of an Ainu - for instance, Melkor, watching from the shadows - they would have glowed with the light of the Two Trees, a stark contrast with their dark kin. And that gave Melkor an idea: to lead his dark, twisted orcs, he would create - or persuade some of his rebel Maiar to become - new lieutenants: creatures with bright, burning fire to cow their minions into obedience, but (unlike those foolish elves) able to turn it off when the need for secrecy arose, to become a shadow in the night.
And so, in mockery of the Eldar of Valinor, Morgoth created the Balrogs.