First of all, I want to congratulate director Peter Jackson, and all the staff and actors who participated in bringing J.R.R. Tolkien's work to the silver screen. I fear that any translation of such a beloved series of books will always disappoint some. I wish that I was not one of them.
The loss of Farmer Maggot, Bill Ferny, and Tom Bombadil. Farmer Maggot got at least a throwaway line, but the interplay around Bill Ferny was missed.
The loss of the Barrow Downs sequence, in which the four hobbits received their special swords, enchanted to harm the Witch-King (and which Merry used at the Battle of the Pelennor Fields).
The casting of Cate Blanchett and Hugo Weaving as Galadriel and Elrond. While they are fine actors, they don't "feel right" as the two most poweful Noldor remaining in Middle-Earth.
The silly fight between Gandalf and Saruman. The influence of Crouching Tiger, Hidden Dragon is obvious and completely out of place.
The replacement of Glorfindel by Arwen.
The loss of the beginning of Gimli's transformation from over-materialistic Dwarf into a better person, when he asks for strands of Galadriel's hair over all the treasure in the world. At least this was in the Extended Edition DVD.
The extended fight between Aragorn and an Uruk-Hai captain.
The extended flight in Moria, followed by the silly jumping about. Do Dwarfs not believe in safety rails, at the very least?
The loss of Gandalf's letter at The Prancing Pony, explaining who Aragorn is, the couplet (originally written by Bilbo), and the fact that The Sword That Was Broken was in Rivendell, and not with Aragorn.
The fact that Saruman is now a direct ally of Sauron from the get-go, as opposed to his trying to gain The Ring for himself to challenge Sauron for mastery (which becomes undone after he loses his army at Helm's Deep).
The slip-and-fall by Frodo on Caradhras in which the Ring manages to slip off from Frodo's neck, for Boromir to pick up.
The ridiculous break in continuity when the Ring, which Frodo has worn on a chain, mysteriously comes off, flies through the air, and conveniently lands on Frodo's outstretched finger.
The Fellowship of the Ring / seiyuu.com!okamoto / 13 January 2003