LIMBO! That game rules and has the belakest atmosphere of all.
Also, although its not really a "GAME" though i still strongly urge you to try it.... a piece called "Dear Esther"
The game is called Limbo, its available on ps3, xbox 360 and pc via steam.
Here is a direct link http://limbogame.org/
I can only hope many more buy this beautiful game, the wife and I just recently played through it together. Its terrifying, amazing and beautiful all the same time. It gave us both some genuine scares, extremely captivating.
In the story-driven, nice soundtrack platform games: Braid and Limbo are both awesome.
The Swapper also has a great atmosphere and a pretty decent story.
Sadly it is not available on Android yet. The team has posted a job opening for a android developer to port it, but it does not look like it has been filled yet.
For now you can get it on a bunch of other platforms though (Xbox 360, PS3, Windows, OSX and iOS + Vita (Thanks /u/stbilyumchill)): http://limbogame.org
Gizmo's and Gadgets - old school game where you had to travel around a warehouse solving science/logic puzzles to create the best machine for a race.
Limbo - a game about death and tells a story through imagery, also contains logic puzzles
More a wry commentary on the nature of proof as to what happens when you die, as we seem to have an alarming lack of credible testimony from dead people. There is a Limbo and it's not a bad game. Bit short, but clever.
this might be a bit too tangential, but you might want to check out Limbo too. It doesn't even use color, and it relies on gameplay deaths and minimal audio to tell you about the world.
Bastion (http://supergiantgames.com/index.php/media/)
Superbrothers: Sword & Sworcery Ep (http://www.swordandsworcery.com/)
Limbo (http://limbogame.org/)
I believe each of these are $5.
I just finished Limbo and The Stanley Parable not too long ago. Unfortunately, they're both the types of games that you just do not want to watch videos or read articles about. The reason is that doing so will likely ruin the experience of playing them. You'll understand when you play them. Fortunately, The Stanley Parable is totally free, and Limbo has a free demo on Steam so there's no reason not to at least try them. They're also very short. The Stanley Parable takes about 10 minutes to play, and Limbo takes about 3-5 hours.
Without giving away too much, I can tell you that both games were awesome and rather artistic, though in different ways. You said you want a game with an interesting storyline. Playing Limbo and solving the puzzles are only part of its experience; the rest is trying to interpret its meaning. In contrast, The Stanley Parable consists of multiple stories that plays with the psychology of the player.