Touch posted...
You mean Peter Jackson lied to me AGAIN
This is actually one of the longest-running debates in LotR fandom. The problem is that there's two paragraphs - both almost back-to-back - that seem to provide conflicting information.
The first passage describes the shadows gathered around it "like two vast wings", suggesting that the balrog actually has no wings but kind of looks like it does; the second passage (taken from when it actually confronts Gandalf) includes the line "its wings were spread from wall to wall", which suggests that it does have wings. This has led to a number of interpretations:
1) It has shadows that look like wings, but aren't.
2) It has wings, but they are covered in shadow most of the time.
3) It has control of the shadows that surround it and can form them into wings, or at least wing-like shapes.
And it's maddeningly unclear which one Tolkien was going for, because circumstantial evidence abounds.
-Balrogs are never depicted as unambiguously flying (one passage does describe Balrogs as "flying" from Morgoth's fortress of Angband, which was located in a mountain range suitable for flying, but context suggests it was almost certainly intended to be "fleeing" rather than in-air flight, the same way Gandalf says "Fly, you fools!" and doesn't seem to intend for Boromir and Aragorn to start flapping their arms and take to the air). However, it's worth keeping in mind that wings do not necessarily translate to a capability for flight, as birds like the ostrich and the penguin can helpfully demonstrate.
-Two balrogs are depicted as falling from great heights - Durin's Bane, the one fought by Gandalf in Moria, and one named Gothmog, killed by Glorfindel during the fall of Gondolin. That said, both balrogs were engaged in combat with powerful foes at the time, which may have hindered any ability for flight they once had.
-Speaking of Gondolin, the balrogs got there by riding on the backs of dragons, which would be odd to do if they could just fly there themselves. That said, it's not proof of anything - humans can walk and run just fine, yet we still rode horses to battle for thousands of years.
-Finally, there is a passage where a Nazgul flies overhead on a fell beast and Gimli comments that the shadow reminded him of the balrog, as though he expected it to be one. That would be an odd expectation if the balrog didn't have wings and/or was inpacable of flight.
Reading between the lines, it *seems* that balrogs
probably
didn't have actual, functional wings, but
probably
had something that looks a lot like them. Most visual depictions of balrogs have wings simply because wings look really cool on a fire-and-shadow demon.