Minecraft is an exploration- and survival-based sandbox game in an almost endless procedurally generated world. The player is thrust into the ground and given until nightfall to provide himself with enough shelter to survive the onslaught of monsters that ensues in darkness.
There is no right or wrong way to play Minecraft. Whether you turn the difficulty up to hard and try your best to survive the night or lower it down to peaceful and just have a leisurely time constructing a whole city, there's always something to do. There are two types of gameplay, trying to survive in Survival, or to simply build freely in Creative. The game does, however, have a proper end, if that's your sort of thing.
The full version of Minecraft was released on the 18th of November, 2011 following nearly 2.5 years of open alpha and beta releases. The full version is still in active development for both new features and bug fixes. To follow its development you can read the Mojang team's blog. The game is currently being written mainly by Jens Bergensten and Nathan Adams, with assistance from the other developers at Mojang.