Userscripts and Firefox
Greasemonkey is an extension for Mozilla Firefox, an open source Web Browser. Most userscripts are written for Firefox & Greasemonkey (although some work in Opera, Safari and even Internet Explorer).
For this guide I will assume you are using Firefox, if not you should install Firefox first.
Userscripts run via Greasemonkey
Now that you have Firefox, you need to install Greasemonkey. After installation (which requires restarting your browser), you are now ready to install userscripts.
Now clicking on a .user.js link triggers Greasemonkey to pop up the script installation panel. Greasemonkey shows you a list of what sites the script will run on and ask if you want to install the script.
Now loading a web page results in additional code (the userscript) being run.