How did it happen in the movie?
Before reaching the western entrance to the mines, Gandalf has a small chat with Frodo, telling him to be very careful because danger threatens his quest from the outside, but also from within.
When they finally come to the Walls of Moria, the passage hindered by a great dark lake, they find no door at first, only two holly trees. Gimli explains that the doors to Dwarven strongholds are often lost, because they are hidden so well, which earns him a sarcastic remark from Legolas.
A little later, however, the moonlight hits a portion of the wall and suddenly a door lights up. The inscription reads: ‘Speak friend, and enter.’ Gandalf tries a few passwords and spells, but fails to find the correct one until Frodo solves the riddle by asking what the elvish word is for ‘friend’.
When the Fellowship enters the Mines, they find the floor strewn with skeletons of Dwarves, and Orc arrows. Upon leaving the entrance, however, Frodo is dragged into the water by one tentacle. When Boromir and Aragorn chop the arms off, more arms become visible – arms belonging to a hideous monster that apparently resides in the lake.
The Fellowship is forced to return into the Mines and travel through Moria, when the entrance collapses and blocks their way out.

How did it happen in the book?
The Fellowship journeys to the West-gate of Moria with Gandalf as their guide. They find their passage hindered by a huge lake, where all the water of the mountain-stream Sirannon has gathered. When Sam leads Bill the Pony through a shallow stream, the companions become aware of a disturbance in the water.
At the Gate, Sam is told to unload Bill and send him away; which only happens after Gandalf has a word with Bill himself. While the Fellowship is packing and unpacking, so they can travel a little faster through the Mines, Gimli and Gandalf look for the doors.
At first they can’t find anything, but then the moon hits a specific portion of the wall and as if out of nothing a door appears. The inscription reads: ‘Speak friend, and enter.’
Gandalf tries a number of passwords and spells, but fails to find the correct one. Finally it dawns upon him that the correct translation is: ‘Say ‘Friend’ and enter.’
Just as the Gate opens, Frodo is dragged into the water by a snake-like tentacle. All the companions are frozen in shock, but Sam still has enough wits to cut off the arm and drag his master into safety. At that, while the Fellowship is running into the Mines at Gandalf’s command, some twenty slimy tentacles appear.

This scene has undergone a few major changes. Let’s take a closer look at them:
– There is no little chat between Frodo and Gandalf, and Gandalf doesn’t warn Frodo for ‘a threat from inside the Fellowship’ (read: Boromir).
– While Bill the Pony is let go, it’s mostly because Gandalf and Gimli fear the Pony won’t be able to walk up the stairs or climb slopes in the dark. Sam does a great deal of protesting, and is only willing to start unpacking after Gandalf speaks some words to Bill that should guide him to safety. While waiting for Gandalf to open the door, the Fellowship busies itself with unloading Bill and sorting out their belongings to keep only the necessary.
– It’s Merry that gives Gandalf the clue to opening the doors, when he asks what the translation of the inscription – ‘Speak friend, and enter’ – means: Gandalf remembers this after he’s tried every spell he can come up with, and finds it more than a little amusing that it should be Merry who solves the riddle.
– It’s not Merry and Pippin who throw the stones in the lake out of boredom, but Boromir; and it’s Frodo who corrects him and warns him not to disturb the water, not Aragorn. The Ring-bearer grew wary of the water when he had to cross it and felt sudden dread fall upon him.
– Sam saves his master when he is dragged into the water by the tentacle, the others are frozen in shock – so no exiting battle between the Giant Squid From Hell and our three valiant heroes.
– The Fellowship enters the Mines only once, and the owner of said tentacles slams the doors shut behind them and barricades the gate with boulders and the two holly trees that guarded it – or so Gandalf assumes.

Mistakes
– In the books, the doors open outward and can only be opened by hand from within – that is why one needs a password to enter from outside. But in the movie, the doors open inward!

– According to Trignifty, when they enter the mines Gandalf doesn’t have his hat on and when he appears next he does, but there’s not enough time in between these shots.

– There aren’t any dead Dwarves lying about on the floor. Gimli even assumes that Balin never entered Moria, until they find his tomb, because there is no sign of recent Dwarf activity at all.

Borrowed Lines
– When Gandalf has his little chat to Frodo, he tries to prepare the Ring-bearer for the fact that he might not live through this journey. He does so using a line, which he says a lot earlier in the books: ‘There are many powers in the world, for good or for evil. Some are greater than I am. Against some I have not yet been measured. But my time is coming.’ [2.I. MM]

– Gimli’s amazed ‘The Walls of Moria’ is taken from Gandalf’s ‘There are the walls of Moria.’

– Gandalf’s ‘Oh it’s quite simple. If you are a friend you speak the password and the doors will open’ was actually spoken by Gimli in response to Merry’s question about the meaning of the inscription: ‘That is plain enough. If you are a friend, speak the password, and the doors will open, and you can enter.’ [2.IV.JiD]

– Frodo’s comment to Boromir – ‘Don’t disturb it.’ – has moved to Aragorn – ‘Do not disturb the water.’ – who uses it to ‘chide’ Pippin. [2.IV.JiD]

Bookie Details
– The design for the Gate of Moria was taken from the good professor himself, who’s original drawing was also incorporated in the books.
The overall set design was based on a painting by John Howe, which has in turn this drawing as inspiration.
The symbols in the door each have a specific meaning: the hammer, anvil and crown with seven stars are the symbols of Durin, Father of the House to which Gimli belongs and founder of Khazad-dûm (now called Moria); the two trees are the emblems of the High-elves who dwelt in Valinor; and the single star belongs to the House of Fëanor.
The two holly-trees that stand beside it point to the Elves of Eregion, or Hollin as it is called in the common tongue. Celebrimbor, who helped manufacture these doors, was one of the Elves who lived in this land, which borders on the West flank of the Misty Mountains.

– The first spell Gandalf uses, ‘Annon edhellen, edro hi ammen!’, is taken directly from the books. The second spell in the book (‘Fennas nogothrim, lasto beth lammen!’) is replaced by ‘Ando Eldarinwa a lasta quettanya, Fenda Casarinwa!’.

– Sam is the first one to cut off a tentacle. In the books, he’s also the only one.