There is no song that "makes me weep every time I hear it," but there are several songs that "tear at my heart strings," and for different reasons.
Chris Bathgate's Serpentine made both me and my wife cry the first time we heard it together. A confluence of events in our lives coincided with a lovely song:
Shortly after my daughter was born, my wife went to the bathroom to clean up and while she was in there, I had a great moment with my daughter. While just me and my lil' girl were together for the first time, I sang this song to her. When my son was born, I carried on the tradition, as a result this song makes me very happy and brings forth the happy kind of tears:
Recently, as many of you know, my dog died. While I was digging his grave I listened to Beck's new album. I can't describe how sad this was and how much I cried. It was perhaps the saddest and most grief stricken I've ever been. I've lost some humans that were close to me, but nothing touches loosing my pal Hemingway. He was still alive, in pain while I was digging and I recall this song in particular bringing me to tears: