While many of the errors I saw in The Lord of the Rings: Fellowship of the Ring were simply examples of dramatic license, there were some full-fledged, blasphemous errors.

In Galadriel's words at the beginning of the movie (in the book, these words were spoken by Treebeard, but that's not important): the word 'gwilith' (air as a region) was spoken as 'wilith'.

Arwen, in the phrase “Frodo, im Arwen.” addresses Frodo as ‘Frodo’. However, in Sindarin, his name would be Iorhael (or Taur). Even if she were addressing him with the name he would be most familiar with, she should use ‘Maura’ – Frodo’s name in Westron. (Remember that supposedly Lord of the Rings is a translation by Professor Tolkien of the Red Book of Westmarch, written in Westron. Frodo was a native speaker of Westron.

Bilbo’s book of his adventures (what we know as ‘The Hobbit’) was written in English!

Gandalf and Saruman address each other with their Mannish names. However, since they are both Maiar. They would address each other by their names before they came to Middle EarthOlórin and Curumo respectively.

At the gates of Moria, Gandalf, instead of chanting the spell (the one that begins with “Anon edhellen, edro hi ammen…) completely in Sindarin, switches to Quenya in the middle. Perhaps the fact that his Sindarin counterspell was broken by Saruman’s Quenya avalanche spell previously at Caradhras prompted him to make the decision that Quenya would be more effective.

(No, I didn't remember all the dialogues from memory. I consulted Gwaith-i-Phethdain at http://www.elvish.org/gwaith/)

A minor correction to the above (otherwise excellent)writeup:

In the book Frodo vanished in a flash of light. In the film, he just disappeared.

OK, so they're rather pedantic. And nitpicky. I'm not attacking the movie here; as a matter of fact I'll be the first to tell you that it was very well done. I just tried compiling as many errors as possible for fun.)