Arwen and Aragorn: Mostly just 'cause I'm a sucker for cheesy/ beautiful romantic scenes, but there is something compelling about this relationship. I suppose because it's timeless, full of hope in times of despair and because it's desperately tragic. *cries then starts singing*
someday my prince will come! 
I agree with Bungo on Merry and Pippin's relationship. Billy Boyd and Dominic Monaghan did an incredible job lightening up the mood and really showing a true friendship on screen and off.
Galadriel and Celeborn: I like their relationship in the books and in the movies, I think cate blanchett was an awesome Galadriel (but that's probably for another forum). The way that it played in the movie was Celeborn said something and Galadriel gently correcting it or, no, that's incorrectly worded adding more maybe so that it seemed that she was the stronger of the two. Which was smart of PJ and tolkien to do because there are no really other feminine characters besides eowyn. I thought it was especially smart for PJ because he insulted Eowyn at times and told her she wasn't strong enough to fight (not him personally, through the form of eomer in ROTK:EE). It was good to have a feminist spin on things in an otherwise masculine movie.