Characteristics
Tolkien's conception of Balrogs changed over time. In all his early writing, they are numerous. A host of a thousand of them is mentioned in the Quenta Silmarillion, while at the storming of Gondolin Balrogs in the hundreds ride on the backs of the Dragons. They are roughly of twice human size, and were occasionally killed in battle by Elves and Men. They were fierce demons, associated with fire, armed with fiery whips of many thongs and claws like steel, and Morgoth delighted in using them to torture his captives. They were loyal to Morgoth, and once came out of hiding to save him from capture.
In the published version of The Lord of the Rings, however, Balrogs became altogether more sinister and more powerful. Christopher Tolkien notes the difference, saying that in earlier versions they were "less terrible and certainly more destructible". He quotes a very late margin note that was not incorporated into the text saying "at most seven" ever existed; though in the Annals of Aman, written as late as 1958, after the publication of The Lord of the Rings, Melkor still commands "a host of Balrogs". In later writings they ceased to be creatures, but are instead Maiar, lesser Ainur like Gandalf or Sauron, spirits of fire whom Melkor had corrupted before the creation of the World. Power of the order of Gandalf's was necessary to destroy them, and as Maiar, only their physical forms could be destroyed.
Tolkien says of the Valar (including the Maiar) that they can change their shape at will, and move unclad in the raiment of the world, meaning invisible and without form. But it seems that Morgoth, Sauron, and their associated Maiar could lose this ability: Morgoth, for example, was unable to heal his burns from the Silmarils or wounds from Fingolfin and Thorondor; and Sauron lost his ability to assume a fair-seeming form after his physical body was destroyed in the downfall of Númenor.
Tolkien does not address this specifically for Balrogs. In "the Bridge of Khazad-dûm" in The Fellowship of the Ring, the Balrog appears "like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater". Though previously the Balrog had entered the "large square chamber" of Mazarbul, at the Bridge of Khazad-dûm it "drew itself to a great height, and its wings spread from wall to wall" in what was a vast hall.
The Balrog's size and shape, therefore, are not given precisely. When Gandalf threw it from the peak of Zirakzigil, the Balrog "broke the mountain-side where he smote it in his ruin".
Whether Balrogs have wings (and if so, whether they can fly) is unclear. This is due partly to Tolkien's changing conception of Balrogs, but mostly to his imprecise but suggestive and possibly figurative description of the Balrog that confronted Gandalf in Moria.
The two key quotations:
His enemy halted again, facing him, and the shadow about it reached out like two vast wings.
… suddenly it drew itself up to a great height, and its wings were spread from wall to wall …
The argument hinges on whether the "wings" are physical wings or simply figurative wings of shadow. Many additional facts are adduced to the argument, but there is not enough firm description in Tolkien's writing to settle the argument definitively.
The Balrog of Moria used a flaming sword ("From out of the shadow a red sword leapt flaming") and the characteristic many-thonged whip of flame in its battle with Gandalf. In The Silmarillion, they also used black axes and maces. Earlier writings also speak of steel claws and iron mail.
In Christopher Tolkien's study of his father's earlier Lord of the Rings drafts, some further indications of Tolkien's evolving conceptions appear. Notably,
A figure strode to the fissure, no more than man-high yet terror seemed to go before it. They could see the furnace-fire of its eyes from afar; its arms were very long; it had a red .
At this writing Tolkien contemplated an edict of the Valar concerning Balrogs, having Gandalf challenge the Balrog by saying:
It is forbidden for any Balrog to come beneath the sky since Fionwe son of Manwe overthrew Thangorodrim.
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