Probably both of them knew what a Balrog looked like. But
The Elf Legolas instantly recognised the Balrog despite having never seen one before. More importantly, the Wizard Gandalf was there (perhaps for this very purpose); knowing that it was far more powerful than even the greatest of his companions, he challenged it.
The elves of Lorien definitely knew that it was a balrog, but so many of them fled the area soon after, I'm not sure how common that knowledge would be there during the fellowship. Galadriel and Celeborn definitely knew, but they apparently never thought it necessary to inform anyone else.
In January of 3019, the Fellowship of the Ring travelled through Moria on the way to Mount Doom, reaching the Bridge of Khazad-dûm in 15 January. There they encountered the terrible creature: the Elf Legolas instantly recognized it was a Balrog, while Gimli dropped his axe in awe, recognizing Durin's Bane.
Of course not. But he knew that there was a nameless Terror that drove out the dwarves, as did everyone else… and so he perceived that Gandalf would naturally be the most likely one to face it.
Therefore, the only explanation as to why Gandalf doesn't reveal the Balrog's presence to the others earlier in the films, is that it is a plot device used by the writers and the creative team in order to increase the dramatic tension of Frodo's decision.
Answer and Explanation: Yes, Sauron knew about the balrog in Moria, as did Saruman, Sauron's lieutenant. However, Sauron did not attempt to recruit the Balrog to his side. The Balrog is a supernatural being who is on a similar level with Sauron.
By my understanding, the Balrogs were seen as undefeatable up until the battle of Gondolin, when up to eight were slain (3 (?) by the House of the Hammer, 4 by Ecthelion & 1 by Glorfindel), depending on the account read. Gandalf obviously also kills Durin's bane in Moria during LOTR, bringing the total up to 9 (?).
The most powerful of all the Balrogs, one of the chief servants of Melkor, who held an authority hardly less than Sauron himself. A wily commander and fearsome fighter, Gothmog was often accompanied by others of his fiery kind, and at least in the Nirnaeth he had a personal guard of dozens of Trolls.
Thrór was murdered as he entered the East-gate of Moria, leading to the War of the Dwarves and Orcs. Even after the end of the War, many Dwarves refused to reclaim Moria, partially because of the Durin's Bane. Years later however, the Dwarves managed to reclaim the Lonely Mountain.
No, Balrogs do not serve Sauron. Similar to Sauron, Balrogs are Maiar, powerful spiritual beings second only to the Valar themselves. They were corrupted by Melkor's evil before the first age and were made into Valaraukar (Demons of Power). They very strength and wills were bound to Melkor.
The Balrog Awakens...
This story conjures images of Dwarves being bathed in a fiery rage as their digging strikes the lair of the dreaded beast, but The Rings of Power took a different path. The legendary Balrog was not awakened by Dwarves or even mining, but by the falling of a leaf.
The movies didn't show it, but the Balrog all but defeated Gandalf in a battle of spells. So, the bottom line is that some brutish beast like Smaug (regardless of his size) wouldn't have been a threat to a Balrog. The only thing that could save Smaug would be his wings -- if he flew away in full retreat.
Legolas doesn't really talk to anyone outside of Aragorn and eventually Gimli, and even then his words are minimal. As a 5,000 year old elf, he can't really relate to the problems and interactions that the “children” of the Fellowship (aka everyone except Gandalf) have.
Gandalf is one of the biggest entities in Middle-earth, being one of the five angelic Maiar sent there during the Third Age. Yet even he quaked in fear at Balrog of Morgoth.
From this point on, up to the year of the War of the Ring in 3018 there is a cohabitation of the Balrog and the orcs for 538 years! Certainly Sauron knew of the existence of the Balrog (especially at the point when his orcs entered Moria), and the Balrog seems to have tolerated the presence of Sauron's orcs.
Gandalf pursued the monster for eight days, until they climbed to the peak of Zirakzigil, where the Balrog was forced to turn and fight, its body erupting into new flame. Here they fought for two days and nights. In the end, the Balrog was defeated and cast down, breaking the mountainside where it fell "in ruin".
Street Fighter X Mega Man. Balrog appears as one of the final bosses on Street Fighter X Mega Man. His weakness is the Mega Buster and the Soul Satellite.
However, they were ambushed by enemies, including a Balrog. Glorfindel fought the Balrog and succeeded in slaying the monster, but was himself mortally wounded. His body was recovered by the great eagle Thorondor and buried under a stone cairn, where afterward grew yellow flowers.
Balrog. Even in the earliest writings, when Balrogs were much, much weaker and more numerous than they are in the conceptions you probably identify with them (as in, what Gandalf fights), Balrogs are stated outright to be more powerful than dragons.
In all of Tolkien's text I've read, there are three people who have killed a Balrog. Ecthellion, Glorfindel and of course Gandalf. All of these fights ended in a draw with all three sacrificing their lives for the deed.
Gothmog (Sindarin IPA: [ˈɡoθmoɡ]) was the first Lord of Balrogs during the First Age, and the greatest Balrog ever to walk Middle-earth.
Peter Jackson certainly makes the assertion in the theatrical and extended editions of the Hobbit movies that Smaug and Sauron are not only well aware of each other, but are planning to ally with each other. Smaug even seems to have detailed knowledge of Sauron's coming plans.
Firstly yes, of course they knew. Balrogs would have been big in the legends of "The Good Old Days" under Morgoth and the figure of fire would be as recognisable to them as a dragon. Legolas easily recognises it, and there's no hint that he actually has seen one before (unclear if he was even born in the First Age).
As a Valar, Morgoth led a whole slew of Balrogs in the First Age, but because Sauron was only a Maia, the Balrog would have had no loyalty to him. He was his own master in Moria, and no Dark Lord was going to ruin that. Thus, if he really wanted to recruit the Balrog, Sauron would have had to dominate it first.