How did it happen in the movie?
After Gandalf realises that Bilbo’s magic ring is indeed the One Ring, he sends Frodo off to Bree. He himself travels to Isengard, where he will seek the advice of Saruman the White, the leader of the Istari.
Saruman, not the most pleasurable man it seems, acts pretty funny when he hears the Ring has been found; and lets Gandalf in on a few secrets about the movements of the Dark Lord Sauron. He shows Gandalf one of the palantíri, and asks him to join him and Sauron. Gandalf respectfully declines, and after a little fight gets himself imprisoned on the roof of Orthanc.

There he sits and watches how the trees are pulled down one by one and the green vale of Isengard is turned into a desolate mining area. He runs into a moth, which he catches and whispers a message to.
Meanwhile, Saruman keeps using the palantír to communicate with Sauron.

During a confrontation between the two wizards on the roof of Orthanc, Gandalf almost falls off – but is rescued by an eagle. Needless to say, Sharkey isn’t exactly thrilled with this turn of events.
But he doesn’t sit around being idle: in the dungeons of Orthanc he breeds a new kind of Orc – a crossbreed between Orcs and Goblin-men. He affectionately calls them his fighting Uruk-hai, and marks them with the white hand of Saruman.

He sends his army of speedy and sun-resistant Uruk-hai out to catch the Ring-bearer, while he himself sits in his tower and occasionally calls off a storm to block the passage of the Fellowship.

How did it happen in the book?
Throughout Fellowship of the Ring, Saruman is only mentioned once: during the council in Rivendell Gandalf explains his delay, and tells of his meeting with Saruman the White.
Some of the Saruman-related scenes are derived from this account, but not all.

Changes from the book to the movie
In the book, Gandalf is travelling through the Shire when he meets Radagast the Brown, another wizard of his order. Radagast was in fact looking for him, with a message from Saruman the White: he is to travel to Isengard as soon as possible, because the Nine have left Minas Morgul. Gandalf asks Radagast, who has a special bond with all the living creatures, to send word to all the birds and beasts to bring information to Orthanc.
When he arrives however, he finds that Saruman is acting very strange and haughty, calling himself Saruman of Many Colours and offering him to join him, so they can keep the Ring for themselves.
Gandalf respectfully declines the offer, and gets imprisoned on the roof of Orthanc. While he sits there, he witnesses the destruction of the green vale of Isengard.
He is saved by Gwaihir, Lord of the Eagles, who was bringing some information to Orthanc as Radagast instructed him to.

Radagast the Brown was omitted, and small homage was paid to him by letting Gandalf whisper something to a moth: Saruman calls Radagast a ‘bird-whisperer’.
There is no funky wizard fight, and Saruman doesn’t exactly offer to get in league with Sauron: it’s his ultimate goal to keep the Ring to himself.

Added scenes
Everything else, is added:
– The spoiling of Isengard, as shown in the movie, is derived from the description Gandalf gives of what he saw while he was imprisoned on the roof of Orthanc.
– The crebain, who in the movie are clearly spies of Saruman, also appear in the books; however, it is not entirely certain whether the beasts are in the service of Saruman or Sauron… or anyone for that matter.
– While in the book the Fellowship can’t agree on what caused the snowstorm on Caradhras (Sauron or the mountain itself); the movie has Saruman calling a storm and creating an avalanche. The purpose of this seems to be getting the Fellowship to take the Gap of Rohan, since Saruman doesn’t think Gandalf will dare to travel through the Mines of Moria.
– The Uruk-hai are not bred by Saruman, nor are they his creation: they were ‘fabricated’ by Sauron. The horde that pursues the Fellowship in the books is a mixture of regular Orcs from Moria and Gondor, and the so-called Uruk-hai.
– Lurtz is a character created by the filmmakers, he is nowhere to be found in the book.

Mistakes
– Saruman isn’t dressed in white in the book: his robe shifts colour when he moves. Saruman calls himself Saruman of Many Colours, because he sees white as a starting point, not the ultimate colour.
– Elwing pointed out that, during the fight, the wizards using their staffs implies that their power is connected to it somehow. When Saruman takes Gandalf’s away, he is helpless and seems to have lost his power. Tolkien doesn’t imply this anywhere, however.
– ‘Uruk’ simply means Orc. Gandalf uses this name in the Chamber of Mazarbul when he describes the horde awaiting them in the hall: ‘black Uruks of Mordor.’ The Uruk-hai shown in the movie (described in the books as larger than regular Orcs, slant-eyed and with a darker skin) were not created by Saruman, but by Sauron.
– The Uruks bear the White Hand of Saruman on their shields, and the S-rune on their helmets. In the movies, the White Hand is the only symbol used.

Borrowed Lines

Bookie Details
– When the Orcs are pulling the trees in the vale down, you can hear them moan. This is not described in the books; however, Tolkien does have a very specific view on nature, in which trees are living creatures with a personality.
– Gandalf is saved by an immense eagle: this is Gwaihir, Lord of the Eagles, who resides in the Misty Mountains and plays a prominent role in The Hobbit.