Working with other evil beings
Balrog of Moria, Balrog of Khazad-dum, Durins' Bane, the Terror, a Balrog (Sindarin), a Valarauko (Quenya)
Apparently it was in appearance as other balrogs, since the elf, Legolas, named it as soon as he saw it. It is referred to as a fiery shadow.
"It was like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater; and a power and terror seemed to be in it and to go before it."
A part of the fearsomeness of the balrog that confronted Gandalf and the Fellowship was that he was difficult to see clearly, as a shadow was around him and possibly of him. This trait could be one reason some of the evil spirits were called shadows, rather than purely the usual reason that "shadow" refers to ghostly beings in general. The terror around it sounds similar to that of the Nazgul.
The core shape seems humanoid - upright body, two arms, two legs, one head since no second was mentioned; multiple heads do not seem to be in Tolkien's writings.
It is capable of a great leap, as it jumped over the fissure that stopped the orcs.
It has a longish mane. "Its streaming mane kindled, and blazed behind it."
Fire seems attracted to the balrog, as flames came up from the fissure and wrapped around it (unless this was part of magic in the fire, to reach up at those above). The fire kindled the mane on the humanoid part of the balrog, which seemed to be no problem, perhaps even welcomed by the balrog. The fire may even have given the balrog energy, showing up as an addition to the fiery part of its nature.
It had hands and arms and a mind capable of handling tool-weapons familiar to the mortal denizens of Middle-earth. "In its right hand was a blade like a stabbing tongue of fire; in its left it held a whip of many thongs." It used its red sword against Gandalf's white Glamdring. The swords seemed to fight both physically and magically, the fires of the two meeting, so that the balrog's sword flew into molten pieces.
It stopped and did a display to try to appear frightening to Gandalf. Its shadow "reached out like two vast wings", it cracked its whip, and "Fire came from its nostrils."
After Gandalf quenched its fire, "it drew itself up to a great height, and its wings were spread from wall to wall". It needed to make a frightening display after having its fire dampened. That was a huge chamber. The shadow reaching out like wings were probably not fleshly wings and the added height was also probably from its shadow self. Physical wings from wall to wall would not have been functional in such "cramped" quarters and would have to remain folded most of the time in twisty tunnels, not habitually used by then for flight. Gandalf broke his own staff, which released a sheet of white fire and broke the bridge against which it was struck, causing the balrog to fall into the fissure. The "wings" could not prevent the fall, if only for fear of breaking them against the pit sides.Or they did not work, maybe from disuse from cave life. If they had existed, the balrog could easily have tangled in them in such a fall.
If balrogs normally had such giant fleshly wings, they should have been mentioned in the earlier books for the awesomeness of them, even non-functional. But if the wings existed at all, they were probably non-functional shadow or even memories of wings from the times of shadow demon attacks on Arien of the Sun. Those shadow demons are not named, but sound suspiciously like the early Balrog Maiar. Manwe and Varda did not require wings to fly, but winged creatures were known so that the Maia could have thought of making their shadows wing-like to attack the new light. For at first Melkor and some of his demons could fly, but their evil caused them to be deprived of this ability. It was long before Melkor could breed winged dragons. Melkor's dragons in the Book of Lost Tales were machines the story line later changing to organic dragons..
The balrog perceived Gandalf and his shutting spell on the stone door when it took hold of the iron ring of that door. It made a counter-spell that caused the door to leave Gandalf's control. Gandalf countered with a word of Command, but the door could not stand up to the powers used and then burst. The wall and roof fell. Either magic or physical strength allowed the balrog to survive the cave-in, so that it continued to pursue the Fellowship after a while.
The terror around it sounds related to that of the Nazgul, even causing people as brave as Legolas to drop his arrow in fear and Gimli to let his axe fall.
It was at least partly magical fire. Gandalf said, "The dark fire will not avail you, flame of Udun. Go back to the Shadow!" Its fire no longer showed but its darkness increased. Fire kindled its mane and seemed natural to it.
Udun, a Sindarin name, was then in Mordor, a deep dale behind the Morannon, where were the tunnels and armories, forts and and towers, watchfires, around which orc-holds clustered; Sauron's armies collected there for the fight against the new King - including an accidental short-term mustering of Sam and Frodo. It was the repaired old Utumno, in Quenya language, Melkor's old underground fortress where he gathered balrogs and may have bred orcs and great spiders, mostly wrecked during the Battle of the Powers. In maps, Udun shows in the upper left corner of the mountains around Mordor. "The Shadow" could refer to Melkor or where Melkor was, the Void, or to the Darkness, or even to its shadow nature.
Working with other evil beings:
The balrog of Moria was in league with the orcs, as they parted to let it pass and followed it across the stone gangway as if in support of its attack on the Fellowship. The orcs beat drums near it, the reason not explained but has shamanistic overtones. Also the orcs and balrogs were working with trolls, as two dropped stone slabs across the fire to allow their passage. The balrog killed out the dwarves from Moria after they disturbed it in the deep place.
The balrog of Moria had a generally humanoid shape, larger and stronger than human. It had a longish mane. It was capable of great leaps. Its core humanoid maned shape was covered in shadow that could stretch out like wings and had fire in its makeup. It could handle a sword and whip. It held the power of terror. It could do spells, such as countering the shutting spell of Gandalf on the door. It had presence of mind enough to swing its whip to catch its opponent, even though falling down a terribly deep hole. It cooperated with or led orcs and trolls against the Fellowship.
Wings look great visually so we will see pictures with winged balrogs, and the suggestion is certainly present. The actual description is so full of shadowy dread that it is hard to see.