I've had so many songs move me that it's hard to keep track of them all. One band that sticks out to me is Flyleaf. I listened to their CD to death when I first got it. All of their songs strike me emotionally in some way.

Also, I think the emotional-resonance with music happens a lot to me because I'm writer, and listening to certain songs puts me in the right mood for a particular story. I can't write without a proper playlist.