Hmm, I want to say FFVI but the story wasn't the thing that moved me as much as the character development.

Actually, no, I think I will go with FFVI. I played it before the internet ruined everything and the twists in that game moved me and I still remember the feeling of playing certain parts the first time.

MS Saga: A new Dawn

Sonic Adventure 2: Battle.

I love RPG's, but their stories arn't really the best. I forgot! My most recent favorite is Heaveny Sword.

Mine? Final fantasy and God of War, and jak.

Man thats a toughie >< let's see,

Metal Gear series, plot twists, great character developement, kinda twisted :D

God of War, GREAT character developement, revenge, sadness, and anger :D Very moving, very believable.

Wild Arms, GREAT music, character developement, simple yet well executed storys.

Final Fantasy 6, 7, 9, 10. I just didn't get into the characters with the other ones,  although I didn't like the characters in 9 they were believable.

Xenosaga, I adored most of the characters on my team, with some pretty solid voice work and a great artistic style the game was more like an interactive movie that delivered on all fronts.

Chrono Trigger, AMAZINGLY believable characters, great villains, wonderfull landscapes. Another masterpiece from Square.

Time Splitters Futer Perfect, the game was just funny, believable characters, natural sounding conversation :D Very pleased ^_^

Uncharted: Drake's Fortune, Classic!!! Amazing character developement :) Very believable dialogue. Just a classic pulp story that delivers :) 

Lufia 2, and Lunar get mentions :D 

That's tough to say. Chrono Trigger was amazingly well woven, but there was a lot of stuff just left hanging.

Twilight Princess will always be big for me. It was good enough that didn't notice how linear the game was until I was just about to beat it.

Baldur's Gate is more for for the compilation of sub stories that you come across that the overall plot of the game. I also don't play without fanmods.

Majora's Mask was really good for the storyline, in many ways, far moreso than Ocarina of Time.

I can't really think of any other's that come close in terms of how good teh storyline itself was.

We have a winner.

Chrono Trigger is miles above everything else for me. So many incredible plot points woven together into an amazing narrative.

Super Mario Galaxy.

NOT.

Zelda OoT
Secret of Mana

I'm still not sure I've enjoyed a video game story since I was a kid, and I went back to the ones I DID enjoy when I was a kid (FF III/VI is a great example, as is FF II) and found them to be terrible... so, I guess my answer is "none?"

Civilization is good, if you consider that a story. Otherwise... nothing comes to mind.