The full version of Minecraft still has no point, per se, as it is a so-called sandbox game. In essence, I would describe it as a first-person fantasy action/adventure/sandbox game. You're presented with a randomly generated world of infinite size that you must explore to gather materials of various kinds. These materials let you build structures to live in and defend from the enemies which come out at night (and later, potentially other players in multiplayer), and also craft various items, weapons and tools. You can hunt animals for food and light clothing, chop down trees to make wooden tools that you can use to mine stone. A stone pickaxe will let you harvest iron if you're lucky to find any while you're spelunking in the enormous, randomly generated cave systems, usually inhabited by various monsters as well as features like underground rivers and lava waterfalls. Don't forget to find some coal near the surface first, so you can craft torches to illuminate the caverns!
One you have your first iron, smelt it in a furnace so you can craft iron armor and sword which will increase your survivability drastically. By now you may be well equipped enough to travel outdoors at night, if you think you can fight off the zombies and skeletons lurking in the dark. If not, all it takes is a few chickens, some flint, and wooden sticks to craft arrows. Once you've killed some spiders to get string, you can craft a bow and use it to fire said arrows, killing enemies at a safe distance.
Then you can start mining for real, with your new, strong iron tools and durable armor. Don't forget to kill some pigs and put the pork in a furnace to make a good meal that will restore your health. That way you can survive after a harsh battle should you discover an underground ruin with monster spawners inside. Keep exploring the cave system and mine tunnels any time you reach a dead end. If you're lucky you can hear the flow of water or lava on the other side of a rock wall and expand your mines deeper into another cave, leading deeper into the bowels of the earth. Eventually you'll strike gold or find extremely rare diamond to upgrade your tools and weapons even further. A diamond will also let you craft a jukebox now!
Between all this, many players enjoy the creative aspect of a sandbox game. All that stone you're digging up while mining can be used to build grand castles, fortresses, underground bases or whatever you want. Wooden huts look very pretty, although don't start a fire too close to them! Dig up sand at a beach and smelt in your furnace to create glass. Harvest reeds and create paper for pages in a book, and craft yourself a bookshelf to stick in your house.
Create grand buildings with doors that can be opened, spiral staircases, and if you're into it, use redstone dust to create circuitry that lets you manipulate doors and mine cart tracks from afar using levers, buttons and pressure plates. Set up elaborate traps to kill monsters threatening your village, or think of dastardly things to do to fellow players once survival multiplayer has been completed.
Build ships and explore the oceans and rivers of your world. Later in the game's development, you can sail between parts of the world and discover dense forests, snowy regions and harsh deserts. Why not set up colonies in each area you discover and live life as a wanderer?
Or, the best thing of all: Completely ignore everything I just said and go about doing exactly anything else. That's what sandboxes are for. And this is the best sandbox I've ever played in. Just remember to never dig down, and don't let creepers sneak up on you at night...