Minecraft is what they call a sandbox game.
The graphics are awful, but that's part of what makes the game great!
So you spawn in a randomly generated world and you can do pretty much whatever you want in term of exploring, mining and crafting.
At night, monsters spawn too so you have to defend yourself and/or create a shelter to protect yourself.
So you explore, get ressources craft stuff, you can even craft electric-powered things!
here's a trailer made by a player :
YouTube - ‪Minecraft Fanmade Trailer v2‬‏
this one is pretty cool too :
YouTube - ‪This Is Minecraft‬‏
the game now is although a little different than shown in these video because Notch (the creator or the game) releases updates quite often and now it's even cooler