Aside from the friendship of Frodo and Sam, the one shared by Legolas and Gimli is probably the most significant in the Lord of the Rings, so much so that you almost can't talk about one of the pair without discussing the other.  Here's some background on both characters and a discussion of why their friendship is so important to the story.  Since I mentioned the Unfinished Tales, here's a box set that contains it and some other stuff you should read: https://www.amazon.com/Tolkien-Fantasy-Reader-Silmarillion-Unfinished/dp/0345466462/ref=sr_1_3?ie=UTF8&qid=1500810177&sr=8-3&keywords=tolkien+unfinished+tales  Or get it by itself: https://www.amazon.com/Unfinished-N%C3%BAmenor-Middle-earth-J-R-R-Tolkien/dp/0544337999/ref=sr_1_1?ie=UTF8&qid=1500810177&sr=8-1&keywords=tolkien+unfinished+tales  Here's the video I did on unsolved mysteries in Lord of the Rings: https://www.youtube.com/watch?v=VE3a9hiEKY8  And for a bit more background information on Elves, see this video: https://www.youtube.com/watch?v=tiBiNWAtsXo