Balrogs are good old Tollers Tolkien's own brand of demons, the Valaraukar, after some linguistic drift. They were Maiar like Sauron, angels who sided with Melkor when he went Morgoth (Elvish for BBEG).
Like their bro Sauron the balrogs aren't so pretty anymore. Typically they manifest as shadow and fire, with whips of flame. Tollers' descriptions of the critters are vague, to the point fans have spent quite some energy arguing whether they actually have wings or not. There used to be armies of them, with some riding goddamn dragons, but most had been defeated by the end of the First Age in the cataclysmic War of Wrath.
As one of the mightiest foes the Fellowship of the Ring faces, and certainly of awesome appearance, the fiery and shadowy demon is copied by many of the later fantasy universes that were inspired by LOTR. Of course, how much originality one can attribute to the concept of a fire+demon creature, in a Christianity influenced Europe, is a question for another debate.
Dungeons & Dragons had balrogs in its first edition, and they still appear as such in the Judges Guild Blackmoor. TSR's lawyers advised them that they were set to get buttraped in court with a big barbed cock made of shadow and flame, and considering this is the Tolkien estate we're talking about, it wouldn't be hyperbole. So Gary filed off the serials by classifying that as a "type VI", with Balor as a suggested name. And this lot have wings and can fly. It all fooled absolutely nobody, but it was enough to get past the lawyers.
Balrogs were actually present in Warhammer Fantasy Battle, specifically in Warhammer Battle Bestiary, where they're described as Greater Daemons associated with fire. It's not clear which God they represent, though it probably would be Khorne or Hashut. They were quickly discontinued for the same reasons described earlier., though Bloodthirsters can be seen as a spiritual successor.