Likewise, story is one of the most important things for me when it comes to games. Skyrim is not a story driven game at all, so don't bother. Mass Effect is an RPG that spans across three games. It's like a TV series, except you control the protagonist.
I think you'd really love these games; Bioware games such as Mass Effect are unique in the sense that they're character driven, and have a unique approach to storytelling. Their games are by far the best at making the player actually feel for other characters. These games have dozens of ethical decisions and hundreds of dialogue options, and each one has a noticeable impact on the game. Other characters are shaped by your choices and how you interact with them, this approach to storytelling needs to be used more often since it showcases how video games can offer a unique storytelling experience, potentially superior to movies and television. Anyway if you get the Mass Effect games, I recommend getting all of the DLC too, but the DLC prices might scare you.
There's also Dragon Age, which is another RPG series made by Bioware (though it has more traditional RPG gameplay which is a bonus for me). It uses the same storytelling approach as Mass Effect, except each game is really its own game while the three Mass Effect games are all part of the same story. I consider Dragon Age: Origins to be the best game ever made... never before have I seen such a huge scale game hit the mark in every area, and have remarkable attention to detail.
The Witcher series are also story driven games, but the first one has such horrible combat mechanics and lacks the cinematic storytelling of the second. It's worth trying but you might not enjoy it, but you'll most likely enjoy The Witcher 2 since it improves gameplay so much.
Other games worth mentioning are Metro 2033 and Metro: Last Light, which are atmospheric survival FPS games. Metro 2033 is loosely based on a novel, and there's a novel being written by Metro: Last Light (and I hear the original author helped write the story of Last Light). The story in these games is very memorable, Last Light has a more complex story than any shooter I've played.
The Penumbra series and Amnesia: The Dark Descent spring to mind as well. These are well-written games with an excellent story and incredible atmosphere. They do an incredible job pulling me into the game and never letting go.
I couldn't help but notice your profile states "Location: Italy". There's an Italian game I discovered not too long ago with a very interesting story, and incredible storytelling. It's called "Anna", you can find it on Steam or Desura. It's a brilliant and very artistic game, and will really test you. Definitely worth a shot.