This question already has an answer here:

In the "Lord of the Rings - The Fellowship of the Ring", while Gandalf the Grey was fighting inflaming demon Balrog, he fell into cavern. So, did he survived the fall or died and reincarnated as "Gandalf the White"? If survived then how does he become "Gandalf the White"?

marked as duplicate by Walt, Chanandler Bong, Community Aug 5 '15 at 6:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.