Advice: The free modes (Minecraft Classic) don't do the actual game justice.

In Minecraft Alpha, it's currently a survival-orentated game-mode, where the idea is to build your stuff, using what you can pull from the land, starting with naught but the shirt on your back and a fist capable of punching a tree into logs.
The catch: When the sun goes down, monsters come out to play. And by "play", I mean, "do their best to try and beat the living spoon outta you, shoot you with arrows, or reduce you to a red stain on the ground; sometimes all three at once".
The hostile monsters (or "Mobs") come in four flavors right now: Spider (pounces at you), Skeleton (Shoots arrows at you), Zombie (Beats you up) and, the dreaded Creeper (Selfdestruct in explosion by contact with player or death by melee attack).
There are two "friendly" mobs, pigs and sheep, which can be harvested for pork and cloth respectively.
Don't like conflict? Set the difficulty to "peaceful", and no hostile mobs will spawn. But, beware the mob that decides it's a good idea to physically push you off ledges.

If you start playing, be careful; it's addictive like crack, only without the track-marks up the wrists. (Can't do anything about the hallucinations of spiders, though)
Best game you can spend 10 Euro on. Ever. Indie Game Of The Year, All Years.