Games like Minecraft are why I love PC games. It is also an example of why, when people talk about PC games dying off, I shrug my shoulders and go play another PC game.
Minecraft is about as open-world a game as you can imagine. You start off by spawning into a 3-D world made of blocks of stuff. This stuff can be anything from water, dirt, gravel, and sand to coal, iron ore, diamonds, or gold. You have to build everything, the game gives you nothing at the start. But by mining different blocks of stuff and combining them in your inventory (actually, a crafting table in your inventory) in different ways, you can combine different sorts of stuff into tools or better items.
For example, you can take sand and coal, put them into a smelter (which you build using a different recipe) and create glass blocks. These glass blocks make great windows for your tower (or pyramid or house or - well, you get the idea).
The beauty of the game is that only your imagination is the limit. Well, and a few laws of physics. Gravity works, for example; you will fall to your death more than a few times. Also, at night monsters spawn (zombies and spiders and I am sure a few other nasties) and will try and kill you, so you will need some shelter in order to live through the night. Of course, you can make weapons to defend yourself but only after you find the appropriate raw materials.
My comments on this game would not complete without acknowledging the old-school, 8-bit graphics. While I like the simplicity of the presentation, others may not like the retro look of the game. Their loss. This game actually looks fantastic - some of the views are awe-inspiring.
Build roads. Explore dark caverns. Create towers. Plant trees. Do a million other things. Just get it and play it.
Free older version is available here. For-pay (9.95 Euros during the Alpha) version available here. The free version is browser-based, the for-pay version can be run in a browser or as a standalone application. Since it is written in Java, it should run on almost any computer.
Oh yeah, one other thing: it supports multi-player. There is a server version available. I have not tried it yet, but I hear that it is pretty cool sharing the world with other people (as long as they aren't assholes :) )
Support an indie developer and buy this. You will not be sorry, except for all the sleep you will lose. I will leave you with a video - enjoy!