The Pokemon Mystery Dungeon series. Every single one of them. I've never met anyone who has played these games who wasn't brought to tears by their respective ends. Nothing else has managed to make me feel this much for non-human characters. Haven't been able to play the 3DS game since it's not released here yet, but I'm trying to kill my expectations; the standards were so incomparably high with the previous games that this'll have to be an enormous drop-off.
Other than that, I'd definitely say Dragon Quest V. Anyone who has played the game knows what part I'm talking about. Unbelievably saddening, especially since it is immediately followed by a really hopeless part of the story. Although I prefer DQIV on the whole, V manages the tragic element considerably better.
I see that a bunch of people have mentioned the Ace Attorney games, but I'm not sure if "emotional" is really the right word there. They affect me quite a bit, but somehow the thrilling aspect seems to be quite a bit more important than the "emotion".
“These are my principles; if you don’t like them, I have others.” – Groucho Marx
