How did it happen in the movie?
After a drinking session in the Green Dragon with his friends, Frodo returns home to find a rather panicky Gandalf, who’d been away to find out more about Bilbo’s old Ring, snooping around his house.
Gandalf then demands to see the Ring, and casts it in the fire. Markings appear, written in the tongue of Mordor: One Ring to rule them all, One Ring to find them, One Ring to bring them all and in the Darkness bind them. It is revealed then to Gandalf that Bilbo’s Ring is the One Ring, made by the Dark Lord Sauron, and the wizard tells Frodo all he needs to know about the plight he’s in.
Frodo then decides, after Gandalf refuses to take the Ring off him, to leave the Shire with it and take it somehwere save.
Sam, who had been ‘cutting the grass under your window, sir’, is caught eavesdropping and forced by Gandalf to join Frodo; while the wizard himself sets out to seek advice from the head of his order, Saruman the White.

How did it happen in the book?
Almost seventeen years after Bilbo‘s sudden disappearance, his cousin Frodo is still master of Bag-end, and life is beautiful. Suddenly one day Gandalf, who had been visiting sporadically up ’till nine years before his return, reappears knocking on the kitchen window. He tells Frodo that the ring he inherited from Bilbo is dangerous, but waits to explain this further ’till the morning.
After a breakfast, Gandalf starts to tell more about the ring: about its powers, the story of how Bilbo found it and how he himself afterwards became suspicious of it.
He puts the jewel to the test by tossing it in the fire, which reveals that carved along the entire length of the band is the Ring-spell, conforming Gandalf’s suspicion that it is the greatest of the Great Rings: Sauron‘s Ruling Ring.
Frodo isn’t very pleased with this revelation, and Gandalf doesn’t help by telling the full story of the forging of the Great Rings and their history; the Battle of the Last Alliance and Isildur‘s fate; the sad history of Sméagol and how he finally lost the Ring to Bilbo.
But the worst is yet to come: it turns out that Gollum made his way to Mordor where he was made to reveal the whereabouts of the Ring’s current owner. In short: ‘something’ will be visiting the Shire… soon.
After Frodo discovers he can’t simply throw the Ring into the fire to melt it, Gandalf convinces him the Ring has to be taken to Mount Doom where it was made and Frodo finally vulonteers to take the Ring out of the Shire.
At that moment, Sam who had spent the entire day trimming the garden is discovered listening by the window and ordered by Gandalf to accompany his master on his journey.

Changes
– There’s no sign of panicky Gandalf in the books: he simply knocks on the kitchen window as he always did, and the two greet as old friends meet. As a matter of fact, he doesn’t tell much of what he’s discovered untill the next morning, only that the ring is ‘dangerous’.
– This scene, obviously, was reduced to the absolute minimum: a lot of the information Gandalf gives Frodo about the Ring was used in The Prologue, but the story of Gollum doesn’t appear on screen until the opening of Return Of the King where we see it as a flashback, and only part of it. There is a reference to it in this sequence though, where we get a glimpse of how he’s being tortured.
– At the beginning of this sequence, we see Gandalf in a stuffy, cluttered library. This is Minas Tirith, where he finds the scroll in which Isildur wrote about the Ring. In the book we don’t get this information before the Council of Elrond, where Gandalf has to give an account of all his doings.
– During this sequence, Gandalf says that ‘the Nine have left Minas Morgul’, and we get a shot of the Black Riders leaving a green gate. In the book, Gandalf doesn’t have this information untill later: after he leaves Frodo (in June, two months after he arrived) he travels about and encounters Radagast, who tells him the Nine have crossed the River and that Saruman wants him to travel to Isengard. So Gandalf doesn’t end up seeking counsel from the other wizard at all!
– Some of the most famous lines of this chapter, dealing with Gollum, pity and fate, were moved to the conversation Frodo and Gandalf have in Moria.
– At the end of the sequence, Frodo and Sam are packed and ready to go. In the book, there’s a good couple of months between the moment where Gandalf leaves Frodo (June) and Frodo’s move to Crickhollow (September).

Mistakes

Borrowed Lines
– As mentioned above, Gandalf’s reading of Isildur’s scroll is taken from 2.II. The Council of Elrond, where it is written as follows: ‘The Great Ring shall go now to be an heirloom of the North Kingdom; but records of it shall be left in Gondor, where also dwell the heirs of Elendil, lest a time come when the memory of these great matters shall grow dim. It was hot when I first took it, hot as a glede, and my hand was scorched, so that I doubt if ever again I shall be free of the pain of it. Yet even as I write it is cooled, and it seemeth to shrink, though it loseth neither its beauty nor its shape. Already the writing upon it, which at first was as clear as red flame, fadeth and is now only barely to be read. It is fashioned in an Elven script of Eregion, for they have no letters in Mordor for such subtle work, but the language is unknown to me. I deem it to be a tongue of the Black Land, since it is foul and uncouth. What evil it saith I do not know, but I trace here a copy of it lest it fade beyond recall. The Ring misseth, maybe, the heat of Sauron’s hand, which was black and yet burned like fire, and so Gil-galad was destroyed; and maybe were the gold made hot again, the writing would be refreshed. But for my part I will risk no hurt to this thing: of all the works of Sauron the only fair. It is precious to me, though I buy it with great pain…’
In the movie, this is reduced to: ‘The year 3434, of the Second Age. Here follows the account of Isildur, High King of Gondor, and the finding of the Ring of Power. It has come to me, the One Ring, which shall be an heirloom of my kingdom. All those who follow in my bloodline shall be bound to its fate, for I will risk no hurt to the Ring. It is precious to me, though I buy it with great pain… The markings upon the band begin to fade. The writing which at first was as clear as red flame, has all but disappeared, a secret now that only fire can tell.’

– The Hobbit that tells a Nazgul off with the brave words: ‘ There’s no Bagginses around here. They’re all up in Hobbiton.’ is probably Farmer Maggot.
When Frodo, Sam and Pippin are on his farm [1.III.], he tells them about a stranger who came by: ‘Be off!’ I said. “There are no Bagginses here. You’re in the wrong part of the Shire. You had better go back west to Hobbiton…”

– When Gandalf leaves the Hobbits behind he says ‘I must see the head of my order.’ This is based on what he says about Saruman in the book: ‘He is the chief of my order and the head of the Council.’ {Note: the Council Gandalf refers to is the White Council.}

Bookie Details
– Before Frodo enters Bag End, we see Sam leave. In the movie it’s suggested that they both returned from the Green Dragon Inn, and though this is not the case of the book Tolkien does write: ‘But that evening, as Sam was walking home and twilight was fading, there came the once familiar tap on the study window.’