First, Gandalf fell down the bridge, then he fell into deep water. the Balrog's fire went out, Gandalf tells Aragorn, Legolas and Gimli later on that now the Balrog was like a snake when they fought on. Finally the Balrog tries to escape from Gandalf, Gandalf runs after him, so he is safed by the Balrog because he doesn't know the way out. They go up the Great Stair and come out at the top of a mountain (was it Caradhras?) where they fight on again. I think the Balrog has got it's fire again. After a long fight, with thunder and firestorms, the Balrog falls down from the mountain-top. Gandalf, too dies because of his wounds.
Now he doesn't tell exactly on which ways he wandered, but he gets to Valinor in the west and obviously meets the Valar (or even Eru, I can't say sorry). After all, he is sent backto Middle-Earth to end his quest in defeating Sauron.
He is no longer Gandalf the Grey, but Gandalf the white, because the treason of Saruman (being the White, head of the Istari and of the White Council) is made clear. Saruman has openly declared war upon Middle-Earth, so he can't be the White anymore. He's evil now. That's why Gandalf becomes The White now. You will read what happens to Saruman later on.
I hope i could answer your question properly.
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Behold the Great Evil!
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