Minecraft is an addictive game that's randomly created for each user. The purpose of the game? To survive. This Machinima tutorial by X will explain the game better than I can, as well as show you how to play if you're interested. I will state that it's addictive - I'm not a big video game player and I spent 3 1/2 hours on it last night in multi player. I'll also say that X was lucky to find coal so easily. My introductory single player round found me scrambling to build a tower out of rock to survive the night, as I couldn't light torches.
As there are no rules or objectives, beyond surviving and mining materials, you can spend your time in the world anyway you'd like. You can even build a USS Enterprise like codbug!