I don't remember ever crying when I was younger due to things happening in the game (usually just becasue I had to turn it off and go to bed), but more recently, usually due to something extremely moving.
Take Paper Mario 2, for example.. *spoilers ahead*
.. I kind of sniffled when I found out about Bobbery's wife, and I remember a tear or two during the final moments of the game, when the Crystal Stars are calling for help.
Other than that, I can't really think of anymore.
Does making Peach cry in Super Princess Peach count? :D