Who is Gollum in the movie?
Gollum is the skinny little ‘thing’ that found the Ring and took it with him to the Misty Mountains where he lost it to Bilbo, was captured by the enemy and tortured in Barad-dûr until he revealed the name of Baggins and later is found out to follow the Fellowship on their quest.

Who was Gollum in the book?
Gollum was once called Sméagol, and part of a family that lived on the banks of the Anduin. There he murdered his friend Déagol for a Ring they found in the riverbed. The Ring made him invisible, but soon he was kicked out of his community and banished. He went to the Misty Mountains, where he – many years later – loses his ‘precious’ to Bilbo Baggins, as is described in The Hobbit.

Changes in Fellowship of the Ring
In the book we are introduced to Gollum in the Shadow Of the Past chapter, where Gandalf tells Frodo (amongst other things) his life story. The next time we hear from him is at the Council of Elrond, where Gandalf reports how he and Aragorn traced Gollum down and interrogated them and Legolas reveals that he came to Rivendell to announce Gollum’s escape from Mirkwood.
Gollum catches up with the Fellowship in Moria and follows them into Lothlórien, but it isn’t until Frodo and Sam one night discover a log following the boats that Aragorn confirms they are being traced.

In the movie, we see Gollum in a couple of brief shots:
– In the Prologue, we see Gollum’s hand when he is holding the Ring (the hand snatching the Ring from the river is Déagol’s, of course) and we get a glimpse of what he looks like on his little rock under the Misty Mountains.
– Frodo only learns from Gandalf that Gollum was tortured by the enemy, and of this event we also get a quick shot, which you can see here.
– His escape from Mirkwood is not mentioned, in fact he is not discussed at the council at all.
– We get a close-up of his eyes (and it’s clear that the CGI work was somewhat changed between ‘Fellowship’ and ‘The Two Towers’ when you compare it to this shot) when Frodo spots him in Moria. In the book Frodo keeps his suspicions a secret until Sam sees him too, but in the movie he asks Gandalf what it is that follows them – which leads up to the discussion about pity and destiny.
– On the extended edition, Gollum makes an extra appearance as the Log-with-eyes which is spotted by Boromir in the Anduin-sequence.

Mistakes
– In Moria, we can hear him rumble ‘Gol-lum’ in his throat. Tolkien describes it more like a cough however, and this description was followed more closely in ‘The Two Towers’.