You start your journey in Minecraft with only your fists and start mining your first resources. Soon you'll build your first small base to be safe from the evil creatures at night. There are several different resources in Minecraft that can be assembled into new blocks, items and technologies.
In itself, Minecraft has no storyline and you can do whatever you want. The absolute endgame, however, is to travel to the so-called "end" and defeat the Enderdragon there. To get there, however, many many hours of fun, joy and despair await you in Minecraft.