I believe the Final Fantasy Tactics game for the PSX has one, if not the best, stories out there today. Luckily I found the entire story someone has graciously typed out for us. Read on, and understand why this game is just so great.
I saw someone point out MGS, which is very commendable of him. Mgs shows a thoughtful and very well made story, along with many cultural references and perspective on politics...As for me, I feel Final Fantasy 7 should be pointed out, but since it's the most talked game ever, I'll go with the amazing Xenosaga trilogy. Xenosaga=Game for friggin' Geniuses