How did it happen in the movie?
After the council and the formation of the Fellowship, Frodo spends some quality time with Bilbo, who gives him two great gifts: the Elf-sword Sting, which glows blue when Orcs come near it, and a mithril coat. When Frodo wants to try the vest on, however, Bilbo catches a glimpse of the Ring and for a moment shifts into something unrecognisable.
Later, we find Aragorn in front of his mother’s grave. Elrond, ever one for looking gloomy and doomy, tells him in politer terms to lay off his daughter and go claim his destiny.
When the Fellowship is ready to leave Rivendell, Elrond speaks some words to them and with Frodo as their guide, the company departs.

How did it happen in the book?
After the council, where Frodo volunteers to take the Ring to Mordor, the company spends long months in Rivendell. During these months, Elrond puts together a Fellowship that has to help Frodo fulfil his task: Gandalf joins the two Hobbits of his own accord and so do Boromir and Aragorn, who are travelling to Minas Tirith by the same route. Legolas and Gimli are chosen as representatives of their people, and because Elrond wants Nine companions – to counter the Nine who serve Barad-dûr – Merry and Pippin are also allowed to join.
The delay is caused by the messengers who were sent out from Rivendell to gather all news of the enemy and make sure the coast is clear. Among these scouts are Aragorn and Elrond’s own sons: Elladan and Elrohir.
While they are waiting, the Hobbits try to study maps and gather some information. Frodo spends quite some time with Bilbo, who gives him his old sword Sting and the mithril coat he got as a gift from the Dwarves after his adventures in the Lonely Mountain. The Shards of Narsil, constantly carried around by Aragorn, are re-forged by the Elves into Andúril, Flame of the West.
When at last the company sets out, it is already December. They take off under cover of night, much to the dismay of Boromir who protests against sneaking out like thieves.

The Formation of the Fellowship
The way the company is put together is quite different from what we see in the movie: the only one pledging any kind of allegiance to Frodo is Sam. Merry and Pippin insist in accompanying their friend, much to Elrond’s dismay. It is Gandalf, who has his own reasons for joining the quest, who insists on taking the two Hobbits along.
Gimli and Legolas, who at this point haven’t done much to draw our attention yet, are added to have representatives of all the Free Peoples; and Aragorn and Boromir are not even real members of the Fellowship, but are travelling together to Minas Tirith.
At their departure, Elrond clarifies the task of the Fellowship: they are to accompany the Ring-bearer as far as they can or want to. No oath is asked from them, and only Gimli insist on giving one but in the end doesn’t.

Bilbo’s Gifts
This scene, save the peek-a-boo moment which was borrowed from the Many Meetings chapter in the book [2.I.], is actually very close to the book, some minor details aside. Bilbo gives Sting to Frodo because Frodo’s own sword, which he got from Tom Bombadil, is broken and was not re-forged by the Elf smiths.
He also asks his cousin to bring him all the news he can, so he can write it down much like he wrote down his own adventures. He also sings Frodo a song.

Gilraen’s Grave
Nowhere in the books does Aragorn visit his mother‘s grave. There is a scene in the Appendices [Appendix A (v)] which describes how Elrond warns his foster-son that Arwen Undómiel, because of her age, her lineage and her importance to her people; is way out of the Man’s league, and would never give her heart to him.
Later on in the same paragraph, after Arwen has turned away from the Twilight to be with Aragorn, Elrond tells him that he is willing to accept that his daughter may play a part in restoring the reign of the Dúnedain, but he adds that only a King will be good enough for his daughter.

Departure of the Fellowship
In the book, the Fellowship leaves at night and in all secrecy – well, as much secrecy as one can expect from a group with a horn-blowing Boromir in their midst. There is a small moment where Sam insists on taking Bill the Pony along, something, which is not explained in the movie at all.
Unlike in the Journey-to-Bree-sequence, nothing substantial was left out, but the timeline was once again altered: the Fellowship leaves in early autumn (which can be concluded from the colours of the trees in the background) and not midwinter.
There’s no reference to Arwen being there when the Fellowship leaves, but Bilbo is present and stands by Frodo’s side.
It’s also in this chapter that Sam remembers he forgot to pack some rope. Reference to this is made in the Many Meetings-sequence.

Mistakes
– After Aragorn and Arwen have plight their troth, Gilraen returns to the land of Eriador, where she was born and stays their till her death in 3007. It doesn’t say anywhere that Gilraen returned to Rivendell to die.
– Bill the Pony keeps re-appearing all the time, it gets a little tiresome of keeping track of him.

Borrowed Lines
– Bilbo’s ‘My old Ring! I should very much like to hold it again, one last time.’ when Frodo starts to take his shirt off to try the mithril vest on, echoes the line Bilbo says in the book during the feast in the Halls of Fire: ‘I should very much like just to peep at it again.’ [2.I.]

Bookie Details
– For those thinking the special-effects on Bilbo when he sees the Ring are a little over the top, this is what the professor wrote about this moment: ‘To [Frodo’s] distress and amazement he found that he was no longer looking at Bilbo; a shadow seemed to have fallen between them, and through it he found himself eyeing a little wrinkled creature with a hungry face and bony groping hands.’ [2.I.]
– The mithril vest Bilbo hands to Frodo is described as ‘close-woven of many of many rings, as supple almost as linnen, cold as ice, and harder than steel. It shone like moonlit silver, and was studded with white gems. With it was a belt of pearl and chrystal.’ [2.III] Save for the belt, the costume department followed this passage quite closely.
– The inscription on Gilraen’s grave reads ‘Onen i-Estel Edain, ú-chebin estel anim’, which translates out of Sindarin as ‘I gave hope to the Dúnedain, I have kept no hope for myself’, which is a line Gilraen uses in Appendix A (v). We’ll hear this line again in Return Of the King, when Elrond says the first few words and has Aragorn finish it, finally acknowledging his heritage.
– ‘Boromir wore a long sword, in fashion like Andúril but of less lineage, and he bore also a shield and his war-horn.’ [2.III] I’m certain Sean Bean didn’t like the prop department sticking to the books this closely.