The story is all explained with an opening cutscene. A man is shown leaving an apartment and walking across the world. Throughout the world, he encounters many dangerous situations, until eventually he finds the edge of the world. All that surrounds him is complete nothingness. He raises his hand in the air and a brick appears in it, and he starts building.
You play as the man who you name yourself, however his true name is Lucky. You can use the Hand-Pad by touching an icon of a hand on the main screen to summon bricks and other forms of building to build your city/temple/whatever. However, as you progress you do missions to obtain new things to summon, until eventually you can even summon gods from all sorts of beliefs (Thor included!) and weapons of mass destruction. Interactions between different summons and the confusing trials of building stuff exactly how you would in real life, such as filling dredged ditches with cement, really help the game progress. You can not win this game, simply keep building and building and building.
The "Quest" mode is quite a fake name, as their is no quest. You attempt to build a "something", your creations always saved, as you try to learn more summons throughout the game.
Play around with whatever. Your work is not saved from game to game, but you can save things into the Gallery.
Interact with things you built in the Sandbox and share them over local wireless.
Share things you built in the Sandbox with the world and download others creations.