There are no objectives or goals in minecraft yet, but the developers promise for this to change soon.
I do believe, however that this is not necessary. You run around in a randomly generated 3d world and can create and destroy the blocks that this world is made up from. Very easy concept. Apparently in the alpha version you can also build railroad tracks and stuff that actually works. Sweet.
So yeah, all I did the last three hours was run around and build stuff. Check this out:
|First, I built a little temple|
|Then this menacing tower...|
|And then added an underground library.|