This game can no longer be played online. Take a look at
Minecraft Classic is the free, casual version of Minecraft Alpha. You can while away a few hours in a randomly-generated 3D world where you can explore, build and dig to create incredible structures.
While this version doesn't have the survival element of Alpha, the lack of night-time and monsters make Minecraft Classic pleasantly relaxing. You have a huge selections of block with which you can build, from stone to brick, glass, and even TNT. Right-clicking the mouse places a block, left clicking destroys it. In this way to you dig underground, or into mountains, and build whatever structures limited only by your imagination.
There is no way of saving your Minecraft Classic game unfortunately, so your buildings won't last. However, as a way of spending time relaxing in a creative way, Minecraft Classic is hard to beat. If you need more permanent gameplay, check out the paid app here.
Minecraft is a phenomenon, and Minecraft Classic gives you enough to understand why it's such a popular game.