What happened to Gandalf after his battle with the Balrog?

After battling the Balrog for 8 days, Gandalf defeated it, but in so doing also died. Something to note, Gandalf is not a normal person, he is an Istari, a group of being sent by the Valar to assist the peoples of Middle-earth in their battle against Sauron. (The short version is they're supernatural beings sent by higher powers)

By dying when he did, Gandalf had not completed his task of helping to defeat Sauron. In addition, his comrade Saruman had betrayed his order. And so, he was sent back to Middle-earth as Gandalf the White by Eru, the highest power in the Middle-earth universe. He returned 19 days after his death, and then he lay in a trance.

3 days later, the eagle found Gandalf. Lady Galadriel had sent the eagles searching for Gandalf. Upon finding him, the eagle took him to Lorien. There, Gandalf's wounds were treated, and he received his shiny new white robes. He then traveled to Fangorn Forest, where he joined up with Aragorn, Legolas, and Gimli.

This excerpt is from the LoTR wiki:

Gandalf and the Balrog fell for a long time, and Gandalf was burned by the Balrog's fire...Then darkness took Gandalf, and he passed away. His body lay on the peak. The entire battle, from the confrontation on the Bridge of Khazad-dûm to the mutual demise of the Balrog and Gandalf, had taken eight days...Nineteen days later Gandalf returned to life, and lay in a trance...Three days later he was found by the Winglord Gwaihir, Lord of the Eagles, who had been sent by Galadriel to find him. Gandalf was carried to Caras Galadhon in Lothlórien, where he was healed, and clothed in white, and thus became Gandalf the White.

Your question pertains to the movie but the answers are coming from the book. The Two Towers Script doesn't say Gandalf died in so many words. He reports,

GANDALF Darkness took me. And I strayed out of thought and time. [Camera zooms in Gandalf’s eye and enters an amorphous realm of stars and galaxies, ending in a blinding white light] Stars wheeled overhead and everyday was as long as a life-age of the earth. But it was not the end. I felt life in me again.
[Camera zooms out from Gandalf’s eye, lying naked and very still. His hair has turned white and wounds are completely healed. He suddenly shudders with a deep gasp and pants as life returns to him] GANDALF I've been sent back until my task is done.

"and everyday was a life's age on earth" as stated by Gandalf the white upon his return which explains his somewhat foggy memory of his old self. He was 19 days away earth time so using an average lifespan of 70-75 years, Gandalf himself was gone for approximently 1330 to 1475 years (give or take). Given he was Gandalf the Grey for just over 2000 years, it was a considerable time for him personally between his previous incarnation that fought Balrog.