How did it happen in the movie?
After the Fellowship gets stuck in the entrance hall – the one with all the dead Dwarves – Gandalf concludes that they have no other choice but to journey on: a four days walk through the dark shafts and passages of Moria.
We get some great shots of what the Mines look like, and Gandalf gives a little talk about mithril, the foundation of the previous wealth of Khazad-dûm, but also their downfall; and the great worth of, say, a coat of mail wrought in mithril.
After a while, the companions come to a crossing with three gates, and Gandalf has to sit and ponder this for a while. He uses this time to have another chat with Frodo, who has spotted a strange creature following them through the Mines. The wizard identifies it as Gollum, and warns Frodo not to be eager to pass out judgement, because he doesn’t understand the greater design of things.
After concluding that one of the shafts leads to less foul air, the Fellowship travels on and reaches a great dark space: when Gandalf uses his staff to create more light they find a huge hall with many slender dark pillars. In a corner they discern a smaller room, to which Gimli inexplicable sprints. There, in the Chamber of Mazarbul, they discover Balin’s tomb.

How did it happen in the book?
The Fellowship journey, rest, journey and rest. There are two instances where Gandalf is in serious doubt about the right course; and the second time – at a three-forked road – the company spends the night in a small chamber, probably designed for guards who watch the crossing.
Pippin is curiously attracted to the well in the middle of the room, and throws a stone in – the result is a series of tapping signals. Gandalf reprimands the Hobbit and lets him take the first watch, but takes over a little later, sleepless and in need of a smoke.
The next morning he decides to take the right-hand passage, because the air smells less foul there.
At the end of a long walk, the company arrives in a great empty hall, where Gimli sings an old Dwarven hymn and Gandalf tells a few things about the importance and worth of mithril.
Frodo, who’s sight and hearing seem to have sharpened as a side-effect of being stabbed by a Morgul-blade, has had the feeling of being followed throughout the entire journey, and during the rest that night he sees, or dreams that he sees, two luminous eyes approach him.
In the morning, Gandalf suggests that before they plot the rest of their course, they look around a bit to better determine which of the three passages out they should take. They begin with the Chamber of Mazarbul, where they find Balin‘s tomb.

The Rest
The events during the rest at the three-forked road where split in half. The first half takes place here; the second takes place in the Chamber of Mazarbul.
Gandalf is indeed at a loss when they reach the crossing, and asks for a little rest so he can think it through. However, there’s no little chat about Gollum following them or Bilbo being too soft or very wise – this dialogue was borrowed from a much earlier chapter, The Ring Uncloaked.
Frodo ís aware of Gollum in the books, but doesn’t know what it is that’s following them, and doesn’t speak to anyone about it until Sam spots Gollum floating along their boats after they left Lothlórien [2.IX].
Gandalf does eventually decide on their course based on the smell that comes from the entrances.

The Twenty-first Hall
This scene was also cut in three parts. The talk about mithril, which is a little unsettling for Frodo, takes place earlier, when the Fellowship are still travelling through the Mines and stand on the edge of an abandoned shaft.
Gandalf creating some light to show the Hall and look for ways out of it, ís incorporated in the movie, but Gimli’s song and the subsequent rest (with Gollum sneaking up on Frodo) was left out; in stead we see Gimli running to the Chamber of Mazarbul and finding Balin’s tomb.

Mistakes

Borrowed Lines
– Gandalf’s ‘This is the great realm and city of Dwarrowdelf’ is originally said by Gimli. [2.IV]

– Sam’s ‘Now there’s an eye-opener, and no mistake.’ was taken from the scene where the wolves attack the Fellowship: Gandalf uses some magic light to scare the wolves off, and Sam comments on this with: ‘That was an eye-opener, and no mistake!’ [2.IV.]

Bookie Details
– The Great Hall that leads to the Chamber of Mazarbul is described as having ‘ a vast roof far above their heads upheld by many mighty pillars hewn of stone. Before them and on either side stretched a huge empty hall; its black walls, polished and smooth as glass, flashed and glittered.’ [2.IV] The concept art for this scene was based on a painting by Allan Lee.