One of my favourite scenes in the films of Lord of the Rings is Sam's speech towards the end of the second film, where he talks about heroes having the opportunity to turn back. It alludes to the tales in the Silmarillion, Beren and Luthian and others. It's a really nice scene and very poignant.
The other one I like is again at the end of the second film, where Gollum is cursing the hobbits, wringing a branch as though it's their necks. Gollum is gaining control over Smeagol. It's a cracking scene and leads into the music, Gollums song. Brilliant.