In Minecraft, there are blocks, and you can build things with them. It has the same basic appeal as building with Lego blocks, though in Minecraft you have to work to get materials beyond basic wood, dirt, sand and stone, rather than paying by the set.
With pickaxes, shovels, hoes and axes (or, in a pinch, your bare hands) you dig into the earth searching for nuggets of coal, iron, gold and more exotic materials thats the mine part of the title.
At first you use tools constructed from basic materials; later you switch to more durable implements made from sturdier stuff the craft part of the title, though it only begins there.
By combining and refining substances using a crafting table and furnace, you can create sturdier building materials and tools, melt sand into glass, break plants down into dyes, create explosive TNT blocks for demolition, make torches for light, build networks of mine carts and railways, fashion doors, stairways and fences, and forge weapons and armor, among many other items.
Finding those materials takes time and can be tricky, especially when you run into monsters underground thats what the weapons and armor are for or outdoors at night. Some, like Zombies, simply attack you, while the Creeper will explode when it gets close, possibly taking a large chunk of whatever structure youre working on with it watch out for these menaces, but be sure to take them out from time to time they drop gunpowder, and some other monsters drop resources as well.
If the monsters kill you, youll reappear where you originally entered the world, or at the last bed you slept in (be sure to get back to your place of death quickly everything you were carrying will still be there, but not for long). If you play on the easiest setting, the monsters arent a threat.
Its up to you what to do with the tools the game offers. Minecraft is a sandbox; you decide how best to play in it or join someone elses world via the online multiplayer mode.
Of special note: The Xbox 360 version of the game is behind the PC version at the moment, and the map size is smaller, but the developer of this edition has promised updates to the game.