Yeah, different songs make me cry depending on my mood and the circumstances. Some pretty consistant ones are:
"Baby Mine" by several people but it's originally from Dumbo, I think. It'd get stuck in my head when my cat was particularly sick, so it's always got that association now.
"Hold On" by Sarah McLachlan, the 'Freedom Sessions' version with just piano. Gets me every time.
"Good Enough" by Sarah McLachlan, from the same album. In fact, that whole whole album is pretty melancholy.
"Me and a Gun" by Tori Amos. Incredibly powerful song.
"1000 Oceans" by Tori Amos. Again, incredibly powerful song.
And a bunch of classical pieces, none of which I can currently think of 'cause I've got other music on. But man, nothing affects people quite like music.