In high school a friend and I made a "suicide mix", which was really insensitive I know now having had suicide touch my life in several ways. But anyway it was fun to just make a mix of the saddest songs we could imagine, and sometimes those weren't songs you'd normally peg.
I can't find a good youtube (that's not a shitty cover by some douchebag in his bedroom) but one of these is "I Don't Believe You (She Acts Like We Never Met)" by Bob Dylan.
Of course I can't leave out my favorite band, Bright Eyes. Two songs by Bright Eyes (although half of their catalog is sad): "Motion Sickness" and "A Celebration Upon Completion". Also the song "Fond Farewell" by Elliott Smith. Of course this one is personal to me because my childhood friend died of a heroin overdose a couple years ago and this song reminds me of him.