First time I ever cried at a video game was when I was watching my Dad play Tomb Raider and he got killed in a spike pit. Think that was the first time I'd ever actually seen a video game character die. I was really little at the time.
I can remember crying at several stages in the Metal Gear Solid series, but the most powerful was Sniper Wolf's monologue before her death. I also cried when Meryl got shot and you couldn't do anything to help her. Also, Emma's death nearly reduced me to tears, but it was probably only because I felt so sympathetic towards Otacon. I really love that guy.
I always enjoy the storylines to the Final Fantasy series, but the only tear-jerking moments in it for me were in FFVIII, when Rinoa is in a coma, or when he dives out into space to rescue her - and in FFX, when Kimahri learns that his entire Clan is dead.