Alright, so emotionally moving video games has given me many choices to pick from, so how about...

Poets of the Fall - Late Goodbye from Max Payne 2

Iji - Further
It's a really good cover, which sounds even better after finishing the game.

no lirycs in this one but this is the best in my opinion
http://www.youtube.com/watch?v=6LB7LZZGpkw&feature=my_liked_videos&list=LLdXU-w3Dxtln-74DnEk6ITg
the game itself is also good

Final Fantasy 8 - The man with the machine gun

There, now stop being emo

There's a load of games with epic soundtracks. But if you want something with emotional vibe I would say:
Katawa Shoujo OST - it's good on it's own but after you beat the game... just WOW
Earth 2160 Intro
and its lyrics(description)
Earth 2160 Outro if you feeling down by previous
Portal 2 - The National - Exile Vilify
These are just the ones from top of my head but there are plenty more. If I remember something I'll reply here

Deus Ex main theme

Mass Effect ending (part2(faunts) by Jack Wall and Sam Hulick)

Kingdom Hearts II's Passion (opening version) not the original but great version

all of Journey's OST by Austin Wintory

ICO's Castle in the Mist

...

No one yet mentioned 'Rebuilt Jedi Enclave' from Knights of the Old Republic 2? I haven't played many games, but of those I played, this is the one track I remember being most moving, especially because of the setting and dialogue of that scene o.O Or 'Suicide Mission' from Mass Effect 2

Sorceress Town theme from HOMM 2

Die Toren sprechen in ihrem Herzen: "Es ist kein Gott, es ist kein Gott."

I feel like we are doing a school project for you.

pokémoooooooon

PlanetSide main theme
Eve online - Red glowing dust

With lyrics:
Small two of pieces - Xenogears
Suteki da ne - Final Fantasy X
Everything's Alright - To the Moon
Dreaming My Way Home - Rogue Galaxy
Simple and Clean - Kingdom Hearts (Japanese version is Hikari)
Sanctuary - Kingdom Hearts 2 (Japanese version is Passion)
Utada Hikaru does both English and Japanese versions of the Kingdom Hearts songs. Her music in general is worth listening to.

Without lyrics:
For River - To the Moon
Overcome the Sorrow - Rogue Galaxy
Dearly Beloved - Kingdom Hearts
Living with Determination - Persona 3
Reminiscence Case DL6 - Phoenix Wright/Gyakuten Saiban series
Emiya - Fate/Stay Night (game version)
Amber Sunset - Hotel Dusk: Room 215
Gyakuten Sisters' Ballad - Phoenix Wright/Gyakuten Saiban series
The Guitar's Serenade - Apollo Justice
Recollection ~ A Fate Smeared by Tricks and Gadgets - Apollo Justice

I'm not sure if anyone had said this before but End of the battle from Shadow of the Colossus

dodododododo---dododododo-dododododododododod-düdüdüdüdüdüdüdüdüüü-dodododooo-düdüdüüüüü-tamtamtamtamtaaaan-tamtamtamtamtaaaan-tamtamtamtamtaaaan-tamtamtamtaaaaaaaaaaaaaaaam-

Legend of Mana, the Jumi storyline. Tears. Every. Bloody. Time.

Any song from Deus Ex: Human Revolution.
Lyrics: HEYYYYYYYYYYYYEIIIIIIIIIIIIIISAOOOOOOOOOOOOOOOOOOOOOOHOOOOOOOOOOOOOOOOOOOOOHHHHHHSOOOOOOOOOOOOOOOOOOOOOOOOHEEEEEEEEEAEIIIIIIIIIIIIIIIIISSSSSSSSSSS