When you first play Minecraft, you find yourself in a natural area (blue sky, green grass, trees, water). Your character can break blocks of wood, and dig up blocks of dirt. You soon realize that blocks can be collected. You soon realize that these blocks can be built into other things, which in turn can be used to build yet more things. Finally, you find that night time is fast approaching, and you must seek some shelter. If you are up and about at night, a monster may come to attack you. Or not.
In Minecraft, you can turn off the monsters from appearing. You can, in fact, enter a "creative" mode where you can call up any particular block that you want, and with this capability you can build practically anything. (The blocks in Minecraft are like Lego blocks, and unsurprisingly Minecraft and Lego are building off that similarity.)
There's only one "rule" in Minecraft: you can only sleep at night. Other than that, you can do practically anything that you want. You can run a large farm, you can grow chickens, you can build a boat and travel the ocean, or you can climb the highest mountain, or explore the deepest mine. You can make maps, write in books, make a large railway or dig for gold.
I enjoy exploring Minecraft's seemingly endless world. Unlike most video games that have pre-defined maps or territories, Minecraft generates its world as you enter them. When your character reaches an area you haven't visited yet, the game produces a new territory for you to explore. It can be a desert, a wintry forest, or a vast plains area. It can be rainy, or snowy.
It's a game that caters to your imagination. I love walking around in Minecraft, getting lost, and building small areas so that I can sleep at night. I pretend I'm "the last person on Earth", except I often play with my daughter on our local server. I love exploring abandoned mine shafts, and climbing mountains. I pretend I'm an intrepid explorer. There's no time limit, so I don't have to rush anything.
Most video games are goal-oriented, but in Minecraft, the only goals are those that you set for yourself. Because of this, everyone's attachment to the game becomes deeply personal. Not everyone plays it the same way, and as a result the game becomes a reflection of you instead of a reflection of its creator. Think of Minecraft like a musical instrument: everyone knows the notes, but everyone plays them differently.
To say that it's my favorite video game is an understatement: it's one of the best things in my life.
