Greasemonkey is a software implementation of a concept called "Active Browsing". It allows users of the Mozilla Firefox web browser to install user scripts that modify specific web pages. Without an extension like Greasemonkey, this modification has to be done manually, using Firefox's Javascript Console or the Javascript: protocol, the browser's Document Object Model Inspector, or bookmarklets. As the Greasemonkey scripts are persistent, the changes made to the web pages are executed every time the page is opened, making them effectively permanent for the user running the script.
Related: Firefox/Extensions#Greasemonkey