When you have bought the game, you can download the stand-alone launcher for Minecraft here. It will automatically update the game files with the latest version. You'll be able to play the game without an Internet connection if you've been online at least once.

Minecraft for Windows

Download Minecraft.exe (270 KB). Place it anywhere you want, then run it.

Minecraft for Mac OS X

Download (270 KB). Unpack and place Minecraft in your /Applications folder.

Minecraft for Linux / Other

Download minecraft.jar (89 KB). The jar is executable and might work as-is. If you run into memory issues, try launching it with java -Xmx1024M -Xms512M -cp Minecraft.jar net.minecraft.LauncherFrame, also please use Sun's JVM.

Multiplayer Server

If you're running on Windows and just want to set up a server easily, download Minecraft_Server.exe (802 KB) and run it.
If you want to run the server on any other OS or without GUI it's a bit more involved (see this wiki article for a tutorial). First make sure you can use java from the command line. On Linux and Mac OS X this should already be set up but on Windows you might have to tinker with the PATH environment variable. Download and run minecraft_server.jar (621 KB) with java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui.