Minecraft for Windows
Download Minecraft.exe (270 KB). Place it anywhere you want, then run it.
Minecraft for Mac OS X
Download Minecraft.zip (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.
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. 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.