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 and run Minecraft.msi. This will install Minecraft and create a shortcut in your start menu. If you'd like a version without an installer, you may use Minecraft.exe instead. You do not need Java installed to run either of these.
Minecraft for Windows (alternative)
If you have issues with the new Windows download, you can try the old one by downloading Minecraft.exe. If given the option, save the file instead of running it. Save the file anywhere you'd like, then double-click the program to run. This does require Java to be installed, however.
Minecraft for Mac OS X
Minecraft for Mac OS X (alternative)
If you are using an older version of Mac OS X and are having issues with the new download, you can try the old one by downloading Minecraft_legacy.dmg.
Minecraft for Linux / Other
Download Minecraft.jar. The jar is executable and should work as-is, also please use Oracle's JVM.
If you're running on Windows and just want to set up a server easily, download minecraft_server.1.8.8.exe 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.1.8.8.jar with java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui.