You start off with nothing in a randomly generated HUGE world (apparently the surface area is about 8 times larger than earth) and your goal is simply to survive. It's a bright day and everything is peaceful, maybe with some cows, chickens and such wandering around. However, at night when it's dark, monsters will come out and it's a bad idea to be out in the open at this time. In order to live through it, you'll have to gather some resources and make a few basic tools, so that you can build a shelter lit by torches (light equals safety). You'll want a safe place that the monsters can't get into. Once you have one and have made more tools (including a weapon) and gathered supplies, it's time to start exploring!
Some worlds are snowy ones. Behold my house built on a cliff by the seaside. On the right you can barely see my greenhouse, growing crops for food.
For me, the appeal is two-fold - creative creation and danger-filled exploration. First of all, everything that you have in the game are things you've made yourself. There's a certain pride and satisfaction you get from that, even in a game. You can build a house (or a fort, or a mansion, or an underground lair with a moat of lava...) for yourself, in any shape you wish. You can use a variety of materials and decorate the place with things like paintings and bookshelves. For example, you can collect sand and clay to make glass for windows and bricks for walls, respectively, by heating them in a furnace. Use your imagination to your heart's content. How about a rollercoaster?
The scary scary dark night. The greenhouse is on the left and on the right is a big lighthouse I built because it seemed like a cool idea. I'm also hoping it will help me find my way back as I explore the surroundings. But mostly I built it just because it seemed like a cool idea.
Actually, there's a nostalgic element to this too. The blocky look of the game reminds me of building secret bases and such out of Lego when I was a kid. Back then I'd always run out of pieces before I was finished, but in Minecraft I can go out into the huge world and find more. Awesome.
Mr Morrgan enjoys the game too. Here's his house by the sea, complete with a manmade stream and planted trees and flowers.
Second, the world is full of cool caverns and cave systems to explore and mine for useful things like iron ore. The deeper you go, the better materials you find for making better quality tools, weapons and armor. It's dangerous, as these are dark places and monsters could be anywhere, but oh so exciting. I have had several great adventures in such places.
And after a scary adventure in the deep, it's a big relief to get back to your house by nightfall. Then you are safe in your home while zombies and other creepy things roam the land outside, and you can fry some bacon for dinner and work on more tools and weapons for your next expedition while you wait for morning.
Darkness falls and the moon rises as I return to my house just in time before the monsters come out. What a welcome sight.
Well, these are just some examples of the things you can do. If you're interested in the game, you can go to the Minecraft website to get the Alpha version that I have (a beta version should be available soon - yes, the game is actually still unfinished). There's also the older but free Classic version you can try out to see if this is your thing.
You can also check out this series of Minecraft gameplay videos to get a better idea of the experience.