Well, to begin, pretty much the entire soundtrack to Angel Beats! can move me and make me happysad. But, the first and foremost song would be Ichiban no Takaramono. Plays at some very very emotional moments. There's two versions, but I'm only going to link one since the other includes spoilers in the video background.
There's also the second ending to Madoka Magica, called Magia, which... Is technically spoilers. But... It's just so unexpected and then awesome at the same time.
There's many more, but... I'll post those some other time.