How did it happen in the movie:
Aragorn, Gimli and Legolas trace Merry and Pippin into the Fangorn Forest. Gimli doesn’t like the forest at all; Legolas tells his companions that it is very old, full of memory and anger, and that the trees are speaking to each other.
Legolas sees the White Wizard approaching, and thinking that it is Saruman, the companions keep their weapons ready. They try to attack the wizard, but he repels Gimli’s throwing axe and Legolas’ arrow with his staff, and Aragorn’s sword blazes and falls from his hands. The wizard, whose face cannot be seen because of the bright light emanating from him, tells the companions that he knows they’re looking for two hobbits, and that those hobbits had been there two days earlier and met someone they didn’t expect. Aragorn tells the wizard to show himself; the light fades and reveals Gandalf. When Legolas apologizes that he had mistaken him for Saruman, Gandalf says that he is Saruman – or Saruman as he should have been.
He then tells the companions – it is shown as a flashback – how he fought the balrog from the deepest dungeon to the highest peak, where he finally managed to overcome and kill it. He himself also fell into darkness, and strayed out of thought and time, where every day was as long as a lifetime. But, he was sent back, until his task would be done. Aragorn addresses him as Gandalf, and only then he seems to remember that name. He says that he’s Gandalf the White now, and has returned “now, at the turn of the tide”.
As the companions walk through Fangorn, Gandalf tells them that they must head to Edoras; Merry and Pippin are rather safe, and their coming there wasn’t an accident, but because of that the Ents will wake up and notice that they are strong.
At the eaves of the forest, where the others had left their horses, Gandalf whistles, and a horse gallops to him. Legolas recognizes it as one of the mearas. Gandalf presents it as Shadowfax, his trusted friend, with whom he has been through a lot. Together, the companions ride away.

How did it happen in the book:
Aragorn, Gimli and Legolas trace Merry and Pippin into Fangorn Forest. Gimli is far from eager to enter the woods, but Legolas tells the others that the forest doesn’t feel evil, there’s just watchfulness and anger, because of some harm the forest has suffered. The forest is also very old, so old that it makes Legolas almost feel young again. From Treebeard’s hill, Legolas sees an old man clad in grey rags approaching; the companions are very suspicious, because the previous night, they had seen a strange old man at their campsite, and right after that their horses had run away. They think that the approaching man is Saruman, and keep their weapons ready.

The old man speaks to them, saying that they probably have already guessed his name, and telling them that the hobbits met someone that they didn’t expect, and are relatively safe now. When the companions see that under the grey cloak the old man is clad in white, they are sure that he is Saruman, and try to attack him. But the man casts away his cloak, lifts his staff, and repels their attack: Gimli’s axe falls from his hand, Aragorn’s sword blazes with fire, and Legolas shoots his arrow high into the air, where it vanishes in a flash of flame. Legolas is the first to recognize the old man as Gandalf, and then the other two also realize who he is. Gandalf himself only now seems to remember the name by which they had called him. He tells them that he is the White now: “Indeed I am Saruman, one might almost say, Saruman as he should have been.”

Gandalf tells the Three Hunters that he knows that Frodo set out for Mordor, and is glad to hear from them that Sam had gone with him. He asks the companions to tell about their journey, and Aragorn tells Gandalf what has happened so far. Gandalf explains to them that it was no accident that Merry and Pippin came to Fangorn, but that they have an important part to play there, and tells about the Ents. He also warns the companions about the Enemy, who already has found out quite a lot about the Fellowship, but fortunately has failed so far – ironically because of Saruman’s actions – and tells them that they must head for Edoras now.

On the way out of the forest, Gandalf tells about his battle against the balrog. They fell long, and Gandalf was burned by the balrog’s fire; finally, they plunged into deep and icy water, down at the uttermost foundations of stone. They fought in dark tunnels gnawed by nameless things that even Sauron doesn’t know. Finally, they came to the Endless Stair that led to Durin’s Tower at the peak of Zirakzigil. There, after a fierce fight, he threw down his enemy. Himself, he fell into darkness, and strayed out of thought and time, and wandered far on roads that he would not tell. But he was sent back for a brief time, until his task was done. Gwaihir the Windlord, the chief of the eagles, found him and brought him to Lothlórien, where he was healed.

At the eaves of Fangorn, Gandalf whistles, and far away they can hear a horse neigh. Soon three horses appear: the runaways, Arod and Hasufel, and a big silvery horse that Gandalf presents as Shadowfax, the chief of the mearas. Together the companions set off for Edoras.

Detail differences:
– In the book, the companions enter Fangorn the day after they met the Riders of Rohan. In the movie, both happen the same day.
– In the book, Gimli sinks on his knees after he recognizes Gandalf. In the movie, Legolas kneels.
– In the book, Gandalf whistles three times calling to Shadowfax; in the movie, he only whistles twice.
– In the book, Shadowfax’s coat is silver; in the movie, the horse is white.

Bookie details:
– In the book, Gandalf tells the others that he was sent back naked. When Gandalf the White returns to life, he doesn’t seem to have any clothes on.