I don't think a game's ever actually made me cry, bar sentimental value making my eyes slightly damp. However, there are games which have affected me emotionally. the following are in ascending order of how much they affected me.
Firstly:
Three words.
"Megszakad a szivem..."
(This breaks my heart... poor Jorge, poor Reach)
Secondly:
In Mass Effect, when my crew was taken, I wanted no more than to go and fuck the Collectors up right that instant. The ship's emptiness hit me hard, dammit - it's always the little, mundane things you miss most, and Bioware deserve credit for realising this.
And finally:
"No-one's left... Everything's gone... Kharak is burning..."
Agnus Dei helped here.