I have several scenes that I think about a lot, and always remind me of the whole meaning of the books, life, and why friendship is the best thing anyone can have.
#1- when Sam thinks Frodo is dead. I cried while I was reading it, and felt that the movie did a good job of protraying the emotions of Sam. This scene had the most emotional response for me.
#2 & #3 tied- When Sam attacks Shelob. That was another powerful example of Frodo and Sam's powerful friendship. When Aragorn leaves Arwen ties with that scene too. Everyone hopes for that kind of intense love (well at least I do), and I think its probably the most beautiful scene in the movie version.
#4 Is definitely the treck through Moria and the climatic ending. I love where Sam is fighting with his pots! And I love how the whole fellowship kicks into high gear when they think Frodo's been mortaly injured. Gandalf's fall is really, really powerful too.
#5 & #6 tied too- I really enjoyed Tom Bombadillo. The part that had me scratchng my head is how the ring had no effect on him. It was a really cool clue as to how and why the ring worked. Also, my probable two favorite foils in ANY book I've read so far, is Gimli and Legolas's first meeting at the Counci of Elrond. Classic scene I think.
Honorable mention- When Aragorn throws Gimli across the chasm to fight for Gondor. Very cool attention to detail there. Average authors would have overlooked the dwarf's inability to jump that far!
And special award for scene just in the movie *drumroll please* is the scene just before the fight for Gondor where Gimli can't see over the wall!! Hilarious. Would quote the movie if I was sure I could convey the feeling.