I want an emotionally moving game.

Well Mass Effect moved me. ME3 made me cry, literally. I hate the minigames the 1st and 2nd have, though.

The void

Rhapsody: A Musical adventure for PSX
http://en.wikipedia.org/wiki/Rhapsody:_A_Musical_Adventure

Deus Ex Human Revolution

Indigo Prophecy

Alice 1,2

Omikron The Nomad Soul

Metal Gear Solid 1,2,3

No idea if anyone said this already, but The World Ends With You, and FFVII in particular made me cry. Don't know why people hate on Aerith, supports are super useful 90% of the time.

Bastion, i cannot stress this enough.

Also, the graveyard is rather interesting, aswell as the path (not exactly "emotionally moving" but they have a huge history background to it)

Heavy Rain seems interesting, considering its from the creators of Indigo Prophecy.

Tales of Abyss (PS2), RPG, starts kinda slow but it's surprisingly good.

Max Payne 1.

Total War Shogun 2. My daimyo's son died... :'(

Braid

Bastion, and the Pokemon Mystery Dungeon series.

The PMD series doesn't look sad, but man, does it hit you at the end.

EDIT: Red Dead Redemption.

My god that has some amazing story.

Slender.

Red Dead Redemption, Heavy Rain, Shadow of the Colossus, Silent Hill 2, FFX

Secret of the Magic Crystals.

hmm.. you may want to steer towards movie license games, ironically, as these generally have real stories with an arc, and thus some kind of emotional attachment, you would hope. Its kind of a reach. ghostbusters, the walking dead. But your question does expose the fact that very few games are emotionally engaging, which is a pretty sad indictment on teh industry. The Dream Machine might interest you a little. Trauma.. but if anything seems a bit disengaging at times... King Arthur 2 (i dont have the first one) has an interesting story progression, though i dont know if i would call it emotionally engaging. games that may be, but i havent actually played: half life 2, deponia, syberia 1/2, cat and the coup.

Threads of Fate, ps1 game. It's a fantasy rpg where you can choose between two character stories that revolve around the same thing. The story is emotionally engaging, the soundtrack is sweet, and the gameplay is pretty awesome. It will make you laugh, cry, smile, and be angry. I guarantee it...sort of. Should definitely check it out.

Final fantasy 7 crsis core for psp ,the ending is so sad. Also red dead redemption has a pretty sad dark ending

Valkyria Chronicles for the PS3 has a really great story. Every Metal Gear game is also amazing. You could also try some Final Fantasy games. I hear FFX is pretty great. If I remember any more I'll add it.

Max Payne 3 - had me throughout the whole story (end was amazing)

It's always like this. There's Shadow of the Colossus but no ICO mentioned by the same guys - Team ICO. ICO is far more emotional than SotC imho. I cried for two weeks (intervals involved of course ;)) after the ending. For PS2 and PS3 the pseudo-HD version.
Chrono Trigger is good as well.

I think I'll get a lot of disagreements with this, but one pretty emotional game was Final Fantasy XIII. Too bad a lot of the haters have never even played through the game once. If you still haven't finished it yet, and have been keeping it on your to-do list, I urge you to start a new game and give it another chance.

Hl2 ep2

You won't be crying. But I found Alan Wake to be an incredibly good story to watch pan out:)

To The Moon

Limbo, maybe not the best if you want tears, but it moves you that's for sure.

Mass Effect series