Hmm...I love music and so many songs evoke so many emotions in me. Off the top of my head, I'd have to list the following as bringing tears to my eyes:
A few by The Cure: Plainsong, Pictures of You and From the Edge of the Deep Green Sea
Sigur Ros: many, many MANY songs by Sigur Ros bring tears to my eyes. He has the voice of an angel...doesn't matter what the lyrics are. Amazing!
Coldplay: The Scientist (like many others have mentioned), A Warning Sign, Fix You and probably Sparks. Oh, and sometimes Trouble. Lots of CP songs make me cry!
Travis: Driftwood, The Humpty Dumpty Love Song, The Cage and Indefinitely. Oh and Pipe Dreams and Sing kinda do too! Lol...
Keane: Bedshaped, Somewhere Only We Know and the remake of Goodbye Yellow Brick Road. Lots of Keane songs make me weepy though haha. Love them!!
Also many Fiona Apple songs - probably Sullen Girl and Oh Well.
Lots of Sarah McLachlan songs, Tori Amos, Regina Spektor, Blue October, and probably a ton more that I'm forgetting at the moment. Oh, well - I love music! I was already planning out the soundtrack to the Twilight saga before SM did so. Hehe...funny - she and I seem to have similar tastes in music! I was pleased to see that she is a huge Muse fan, as am I. Finally got to see them live on their last tour. Many canceled dates before that left me heartbroken! hehe...so I guess Muse makes me cry, too. Lol!