Bioshock 2. I know a lot of people immediately think of the first as having an awesome story, and it does, but I think that Bioshock 2 had a better one. I liked the whole premise of the story: you're a Big Daddy, and you're Little Sister needs you to rescue her. I also liked the way you choices throughout the game determined (SLIGHT SPOILER) how Eleanor reacted to her mother at the end of the game. I thought it was touching (SPOILER END).
Cryostasis. This one is a bit harder to explain. The story itself is fairly simple once the game plays out, but the way it is presented and the way it requires a little thinking to truly understand what happened really made the story stick out. I also liked the atmosphere and the philosophical aspect of the game.
Final Fantasy XIII. I know, I know. It was more of an interactive movie according to some. But I think it has a really awesome story, so there. Sure, it doesn't always make the most sense. But I found it easy to overlook that because of the characters. Now, I've never played any of the other Final Fantasy games, so I can't compare, but even if I had, I don't think my opinion would change. All of the characters seemed real to me. Okay, not completely real (I look at you, Snow), but enough to make me care about them, and enough to make me feel a little bad for Hope at the end.
Oh, and Vanille's my favorite game character ever. Yes, I just said that.
Honorable mentions (but ones I don't feel like writing about right now): Knights of the Old Republic 1 and 2, Deus Ex, Alan Wake, and Max Payne 1 and 2.