How did it happen in the book?
On Frodo and Bilbo’s birthday, Frodo moves out of Bag End. Under the guise of wanting to retire he’s bought a house in Crickhollow, and his friends Peregrin Took, Meriadoc Brandybuck, Folco Boffin and Fredegar Bolger help him with the move.
Merry and Fatty Bolger, as the larger Hobbit is called, travel to Crickhollow with most of Frodo’s possessions while Frodo himself travels with Pippin and his servant Samwise Gamgee on foot.
On the Road they encounter a Black Rider. Unaware of what they’re facing yet, they hide from it. Afterwards, Sam recalls his father told him a similar fellow came up to Bag End the previous day look for a Mr. Baggins.
That same night they are passed by a company of Elves, travelling to the Grey Havens. The leader oft he company, Gildor Inglorion, greets them in a hospitable fashion and the Hobbits are treated to a dinner and the company of the Elves.
When they part, Gildor warns Frodo for the Black Riders but also speaks some encouraging (and prophetic words).

How did it happen in the movie?
Frodo and Sam, at Gandalf’s request, travel from Hobbiton to Bree. On their journey they come across a company of Elves glowing eerily travelling West. Later they encounter Merry and Pippin in a cornfield, and when they finally (quite literally) land on the Road Frodo ‘senses’ a Black Rider approaching and urges them to seek shelter in a hollow under a gigantic tree-root. Frodo feels the call of the Ring, but Sam can prevent him from putting it on and the Rider is delluded.

Mistakes
– Not a mistake, but a scene that is easily misinterpreted: when the Black Rider starts sniffing, bugs and worms crawl from underneath the treeroot. These little monsters are not attracted to the evil of the Ringwraith, as some people think: Peter Jackson wanted to express how all living things fear the wraiths and try to run away.

Bookie Details
– The glow that surrounds the Elves is taken directly from the book: ‘They passed slowly, and the hobbits could see the starlight glimmering on their hair and in their eyes. They bore no lights, yet as they walked a shimmer, like the light of the moon above the rim of the hills before it rises, seemed to fall about their feet.’
– The song the Elves are singing could be O Elbereth! Gilthoniel that is noted down in the book.