I really need to finish To The Moon. I played about 15 minutes of it and got bored, but I've heard such great things about it I feel the need to force myself through it one of these days.

The ending of Final Fantasy X made me cry quite a bit the first couple times I played it. I played it again a couple years ago and even then it still makes me tear up a bit (mostly that part where that sad music plays with all the flashbacks).

Aside of that, I can't recall a game ever making me cry, though the last episode of The Walking Dead: Season 1 did manage to really bum me out.