Bright Eyes is the ultimate "sad music" band. I don't think it's possible to listen to every Bright Eyes song and not shed at least one tear, it doesn't matter who you are.
Duncan Sheik - Little Hands
Brand New - Jaws Theme Swimming
Iron and Wine - Passing Afternoon
Jump, Little Children - Cathedrals
Amos Lee - Colors
Counting Crows - Colorblind
Rasputina - Quitter
As for classical stuff, you can't go wrong with Debussy's Clair de Lune.
Noir OST: Secret Game
Code Geass OST: Stories (can't believe no one mentioned it yet)
Dire Straights: Brothers in Arms (though that is rather subjective)
Fate/Stay Night - This Illusion (piano version)
Adiemus - The Mermaid's Tears
Enya - Exile
I'll edit my post as I'll name the rest. If you like Soundtracks from Animes etc I can give you tons of more sad music from them :)
Kid A - Radiohead
Track 4 - Sigur ros
Morse Code - National Skyline
Breakfast in Bed - Dntel
Light Leaves - WHY?
Broken Drum - Beck
Please sing my spring reverb (B. fleischmann mix) - Mum is good along with their cd
Finally We Are No One