Well, the the moment I can think of two songs:

Simon & Garfunkel - The Sound of Silence

Billie Holiday - Strange Fruit

My uncle.

I get chills every time I hear it.  Even if you don't like country, I'm sure you can appreciate it.

that one goes without saying

less well known, one of many truly epic Oasis b-sides

i get emotional whenever i hear this songhttp://www.mundorare.com/dloads/viva-pinata/Viva%20Pinata%20-%20Main%20theme.mp3

Damn, you beat me to it.

A lot of songs by Pete Yorn have, well...I wouldn't really say they touched me. They're just nice and melodic songs. He's just a friggin' brilliant artist.

Surprise, Soriku posted a videogame song

Many actually, besides most of the ones listed before, Hamburg song by keane. Also, the Aeris theme is pretty sad too.

"One" by U2 brought me to tears the first time I heard it.

"Put That Woman First" by Jaheim seemed like it was written about me and inspired me to get back with my fiance, Maria.

"Here Without You" by Three Doors Down came on the radio right when I was breaking up with a girl I thought I loved. I pulled the car over and cried like a baby.

"Streets of Heaven" by Sharrie Austin -my daughter was diagnosed with a disease back in 2002. She was having seizures several times a day, and we were told that her brain would eventually degenerate to the point where she would become mentally handicapped. This song helped to get me through. Now, almost eight years later, my daughter is beautiful and perfect. She's an "A" student in second grade. She beat Lennox Gastaut Syndrome. Simply amazing.

Other video game songs that are good are Roxas theme from KH2 and Fantasia alla marcha(sp?) from the same.