While the 'lego' comparisons keep coming in, that is the closest the game is to resemblance, its much more than that. Here's what minecraft is:
Classic:
An open canvas filled with scenery already there. You have unlimited blocks to build stuff with and you can change the scenery as you wish. It's all from a first person perspective, and your objective is nothing more than 'go nuts.'
Also, the size of the area is either small (about a mile across) medium (several miles across) and large (pretty huge)
Alpha:
A survival simulator. You are given limited blocks, and must mine and craft the rest. Same as before, you can build and build to your heart's content, but you must mine the materials first, then craft them. At this point, you must survive. Monsters will come to haunt you constantly, and also will try to kill you, but only at night and only in caves. You have health, and if you die, there's little repercussions beyond part of your structure might be damaged, and all the stuff you were currently carrying have been flung around the area in which you died. If you find yourself back to said area, you can pick up all the stuff.
In summary: A building simulator with monsters that hinder your building.
Also, the size of the area is... Unlimited. You heard me. It has no size. Since terrain is randomly generated, the total area of the game is... Infinite. There is a finite area to go up and down in, but beyond that, you can comb the entire land for hours with no barriers in sight.
Beta:
Still in speculation mode, but as far as I can tell, there's going to be a narrative and new 'stuff' to get, craft, mine, and build with.
So! If you're creative and like to make stuff... It's a fun game.