When you have Click "Download Now" , you can download the Beta 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 MinecraftLauncher.jar (1 MB). This is the jar launcher, it contains the version 1.5.2, and it based on the alpha launcher

Download MinecraftLauncher.jar (1018 KB). This is the same launcher but instead its based on Beta launcher.

Minecraft beta 1.8 for Windows

Download Minecraft.rar (6 MB). for legal reasons, this has been removed.

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Multiplayer Beta Server (IN-DEV)

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.