It's like asking "what is Super Mario?".
Minecraft isn't one of those big-budget titles that focus on graphics and story-telling. It's more of a throwback to old-school gaming. The only way to answer "what is Minecraft? " is with "Fun."
Game generates random world, though not "random" random; it makes sense and can be quite beautifull. The world consists of blocks, which you can mine. You can use those to make things; wood makes sticks, sticks and coal make torches, etc. Then throw in a handfull of (nasty) enemies to add a survival component and you've got Minecraft.
Mind you, it's an alpha: plenty of stuff will be added, including a narative to tie it all together and drive a player forward.
Minecraft: Story, Monster Towns, Alchemy