Minecraft for Windows
Download Minecraft.exe. Place it anywhere you want, then run it.
Minecraft for Mac OS X
Minecraft for Linux / Other
Download Minecraft.jar (86 KB). The jar is executable and should work as-is, also please use Oracle’s JVM.
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.
If you want to run the server on any other OS or without GUI it’s a bit more involved. 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 runminecraft_server.1.6.4.jar with java -Xmx1024M -Xms1024M -jar minecraft_server.jar nogui.
Credits: The Minecraft and Minecraft Realms teams