Older versions of Game Maker (6.1A and below) can be downloaded from the old Game Maker Pages. It is recommended that you download the zipped version as some tutorials on this site will assume you have the sprites included in it.
Once you have downloaded Game Maker open the zipped folder and double click on gmaker.exe. This will bring up the setup. Click through it, being careful to read the Licensing Agreement. That's it! You've Installed Game Maker!
Version 7 is the latest version of Game Maker and can be found at the YoYo Games Download Page. It features several improvements over older versions including enhanced security and a new extension mechanism. As of version 7.0 Game Maker is managed by YoYo Games.
CTW Version (5.3)Edit
The Children's Technology Workshop Version of Game Maker is a non-violent version made for the CTW camps. Though still powerful it is a few versions behind the standard edition, is only available as a 15 day trial and lacks any 3D functionality. Most Tutorials should be compatible with the CTW version and sections that are not backward compatible will be clearly marked. The CTW version of Game Maker can be downloaded from Children's Technology Workshop.