How did it happen in the movie?
After the Fellowship leaves the Dimrill Dale, they arrive in a huge mossy forest where an Elf witch is said to be living – Frodo has a close encounter of a telepathic kind with her, before they encounter a real life Elvish guardian and his brothers. They guide the Fellowship through the woods to the Lord and Lady of Lothlórien. During this journey there’s some trouble with the Dwarf and Boromir has a heart to heart with a very ‘down’ Frodo. Haldir, the guardian, shows them Caras Galadhon, a group of golden trees that form the city from which the Lord and Lady govern their wood.

How did it happen in the book?
After the Fellowship leaves the Dimrill Dale, they make for the forest of Lothlórien. Aragorn goes to seek shelter and council there: he was there before, because the Lord and Lady of the land, Celeborn and Galadriel, are the grandparents of his own betrothed Arwen. In fact, it was in these woods where they plight their troth, on Cerin Amroth, as he tells Frodo when they get to that hill.
On their way to Lórien, the Fellowship halts twice: once to allow Gimli to show Mirrormere, an important place in Dwarvish culture, to Frodo and Sam; and once when Aragorn tends to Frodo and Sam’s injuries. They discover the mithril-coat at this point, long after the fight in Mazarbul.
When they reach the wood, they cross the river Nimrodel, and Legolas tells some of the legends about the Galadhrim that our told amongst the Elves of Mirkwood. He also sings the Song of Nimrodel.
When the companions want to try and make a camp in the trees for the night, they encounter Haldir and his brothers, guardians of the forest. After a brief discussion with Legolas, the Fellowship is allowed to sleep in two trees while patrols set out to look for the Orcs pursuing them. During the night, Frodo sees two green lights coming towards him.
The next morning, the Fellowship is to be brought to the Lord and Lady but Gimli the Dwarf is to go blindfold. As a compromise, Aragorn suggests to blindfold the whole company, much to the dislike of Legolas. Halfway the journey, however, the blindfolds are taken off: word has reached the Lord and Lady of Frodo’s task, and so the company can behold the golden mallorn trees of Lothlórien, and standing on Cerin Amroth they see Dol Guldur in the North, Sauron’s abandoned stronghold on the eaves of Mirkwood, and Caras Galadhon, the capital of the Galadhrim. Frodo catches Aragorn as he is looking at a golden flower, elanor, and remembering how he plight his troth with Arwen on that very hill.

Obviously, a lot of this stuff didn’t make the movie:
– After the Fellowship leaves Moria, there is only a little time for some grief, and then nothing happens until the company reaches the wood. Apparently, the original script did have the Orcs coming out of the mountains in pursuit of the companions, and the guardians of Lórien coming to their aid, but this was cut out off the movie entirely. The discovery of the mithril coat was moved to the scene in Balin’s Tomb. And no singing Legolas either.
– The arrival of the Fellowship in Lórien is very different, and they are quickly discovered. Gimli is picked out as the one breathing the loudest, while in the books it is Sam who is mocked by the Elves.
– The discussion in the trees between Haldir and his brothers and Legolas is replaced by a nightly conversation between Haldir and Aragorn, well after the journey into the woods has started. There is no spectacular rope-crossing of the river. Added material contains Gimli’s supposed insult to Haldir, and the little conversation between Boromir and Frodo.
– The Fellowship does see Caras Galadhon, but Cerin Amroth, Arwen and elanor are not mentioned.

Mistakes
– One of the biggest gripes many fans have with the movies, is the lack of mallorn trees in Lothlórien. The only shot coming close to the golden wood Tolkien describes, is this shot of Caras Galadhon.

– When Gimli insults Haldir, he says: Ishkhaqwi ai durugnul. This is supposedly Khuzdul, the language of the Dwarves, meaning: A plague upon the stiff necks of Elves. However, Tolkien didn’t come up with so many Dwarvish words. To my knowledge (please correct me if I’m wrong), the only word in the list that can be translated from Khuzdul is ‘ai’, which means ‘upon’, as is shown in the old battle-cry of the Dwarves, ‘Khazâd ai-menu’, The Dwarves are upon you.

Borrowed Lines
– When he enters the wood, Frodo hears Galadriel’s voice in his head saying: ‘Your coming to us is as the footsteps of doom.’ This was taken from the scene by the mirror, where Galadriel, after telling him about Nenya and her part in the struggle against Sauron, asks him: ‘Do you not see now why your coming is to us as the footstep of Doom?’ [2.VII.MoG]

– Haldir’s joke about Gimli, ‘The Dwarf breathes so loud we could have shot him in the dark’, is actually one of Legolas’ more humorous lines which he says to Sam when they encounter the Elves in the tree: ‘Yes they are Elves, and they say that you breathe so loud that they could shoot you in the dark..’ [2.VI.L]

– Gimli’s ‘Aragorn! These woods are perilous. We should go back.’ echoes Boromir’s discomfort upon entering the forest: ‘But it is perilous.’ [2.VI.L]

– Gimli’s insult to Haldir, Ishkhaqwi ai durugnul (Supposedly: A plague upon the stiff necks of Elves) originally belongs to Aragorn. When Legolas refuses to go blindfolded with the others Aragorn, paraphrasing Legolas’ own comment about Dwarves, cries out: ‘Now let us cry: “A plague upon the stiff necks of Elves!”‘ [2.VI.L]

– When the Fellowship arrives at Cerin Amorth, Haldir says: ‘Caras Galadhon, the heart of Elvendom on earth.’. The original line, spoken by Aragorn, is ‘Here is the heart of Elvendom on earth’. [2.VI.L]

Bookie Details
– It seems, here and at the beginning of TTT, that Mirrormere was added into the background.
– There has been some debate about the strange way Galadriel, Celeborn and Haldir speak. Though slightly annoying according to some, ‘slow speech’ is indeed mentioned in the book. Haldir says, in chapter 2.VI. Lórien: ‘We seldom use any tongue but our own; for we dwell now in the heart of the forest and do not willingly have dealings with any other folk.’