Answer:
After Gandalf fights the Balrog in The Lord of the Rings, he is carried off by an eagle. There is a scene where he is lying down in a white room and only his head and shoulders are visible.
The next time he is seen is in Fangorn forest. He is now recognized as "Gandalf the White". When Aragorn addresses him as Gandalf, he responds with, "Gandalf ... yes, that's what they used to call me. Gandalf the Grey."
He also explains that he was sent back. But who sent him back? Did he die? Where did the eagle carry him?
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.
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.