These are among the most powerful Maiar, and among the first to be corrupted by Morgoth to his service. Apparently the flying breed was a late innovation, in time for the great and last battle of the First Age; and the few who survived and fled to the deepest, darkest places in Middle-earth were of this winged variety.

Gandalf and the other Wizards (evidently being Maiar themselves) were (imo) sent to Middle-earth in part to counter these fell beasties. Sauron was also a Maia, and apparently the most powerful of the lot to side with Morgoth. His powers in the Third Age were not as great as they had been in the Second, nor in the Second as in the First Age.

But as compared to Balrogs, Sauron would have been utterly unbeatable in combat: had Gandalf stood an even chance, he would have confronted Sauron.

In the movie FOTR, the look on Legolas' face when the Balrog was approaching said it all: the ancient enemy of his race's greatest champions was approaching. That had to be something like an elf's worst nightmare.

Fingon killed Gothmog, but also perished: it seems that the few (two ?) times an elf killed a Balrog, it cost him his own life.