Download

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. 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.
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The Minecraft launcher was updated 01 July 2013. To get latest version of the game, make sure that you have downloaded the latest version of the launcher from the link above.

Multiplayer Server

If you're running on Windows and just want to set up a server easily, download minecraft_server.1.8.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.jar with java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui.