A lot of the games I found moving have already been mentioned here. Among my own top are games like Shadow of the Colossus, Mass effect, Prince of Persia (sands of time), and final fantasy 8, which all have already been mentioned. Some which haven't been mentioned yet, but which I found really moving as well are:
Final Fantasy VII Crisis Core (it's actually one of the very few things (including books) that made me cry)
Kingdom Hearts (especially the first one and Birth by Sleep)
Drakan (the second game completely ruined it though...)
Odin Sphere (though not the gameplay itself, all the different characters' perspectives of the whole story is nice)
I tend to agree with Sniperyeti though, at least partially. In a book it's much easier to tell the reader what a character is feeling and why in detail, while in a game it all has to be shown, and so the more subtle an emotion is, the harder it becomes to show ingame. Add to that, that as player you almost always see things from a third person perspective in regard to emotions of any character, you have to read their expressions and from that deduce their emotions yourself. While in books the characters' minds, so to speak, are often open to you and you know their emotions much more directly.
