Gandalf the White and the Grey..

HI

Im new here. Therefore I dont know if this has been up for discussion before, but still... When Gandalf fought the Balrog of Moria and finally defeated him he was on top of the misty mountains and probably pretty exhausted. I dont know the exact phrase but i think it is something like " and then I fell into darkness" Anyway, as I understand it, he was there after found by Gwahir and saved from the top of the mountain and moreover was from there on Gandalf the white. Am i right?
Anyway, I am under the impression that he sort of died there and then was sent back to the world as Gandalf the white. Did he die? Or Maiar maybe cannot die? Is this something that is explained elswhere?