No other medium has the ability to make me feel emotions like video games. The interactivity just makes everything have a greater impact on me. Generally I tear up rather than flat out sob, but the tearing up has happened in a lot of games.
Mother 3, Terranigma, and Persona 4 are probably the games that made the tears flow the most and the hardest.
Terranigma's gotta be the most recent for me (replayed it a couple months back). The first town theme still gets to me a bit...
Illusion of Gaia hit me pretty hard, too, while on the topic of the Soul Blazer games. I really miss the series's strange, poignant storytelling. The grander aspects of its philosophies on reincarnation and spirituality went so well with telling more subtle, human stories about loss, growing, and life continuing on.
A lot have already been mentioned (Mother 3, Final Fantasy XV's last cutscene...), but I think the first for me was Breath of Fire II, but for the life of me I can't remember which part. I just remember being surprised that a game would make me get emotional like that (and, in retrospect, that such an awfully translated one could). Also, Breath of Fire IV's themes of finding beauty in a horrible world and sharing it with others got to me as well, as cheesy as it is.