Cross Posted at eSchoolNews... Greasemonkey is an extension for the Firefox web browser that lets you add scripts to any webpage to change it's behavior when displayed in your browser. As stated on the Greasemonkey page, you can think of it as being similar to using CSS to control a pages style. User scripts let you control any aspect of a webpage's design or interaction.
For example you could write a script that would insert a link to your local library from any page on Amazon.com. Say you are searching for a book title on Amazon. The search pulls up the page with the Amazon content, but it also places on the page a link for that particular title at your local public library. (There are already a couple of examples of this particular use... I'm going to try to create one that will pull content from our local Multnomah County Library...) For a better explanation of Greasemonkey and what it means to the user experience, take a look at Simon Willison's post, Greasemonkey as a lightweight intermediary.
A few that I have been playing with include...
The New York Times Single Page script. It changes the article link from the New York Times page to point to the Single Page Format, rather than the default multipage view.
The Flickr: Photo Page Enhancer inserts a link to the original size jpg on every photo page, and an easy copy/paste textarea with html code for creation of a thumbnail link.
My favorite is the BoingBoing Butler. It displays the BoingBoing.net page with all the sidebar content stripped away.
These and others can be found at Jeremy Dunck's GreaseMonkey User Scripts Wiki