I've used jQuery, prototype, mootools, and dojo and researched some others.
jQuery is my library of choice because it is the best at DOM manipulation and events, and excellent plugin ability. Also has excellent documentation and is the most popular library. Can be used side by side with other libraries.
Sizzle - CSS selector engine. Just finds DOM nodes via css selector syntax and returns them. Used in jQuery, going to be added to Dojo and possibly Prototype. Mootools will not be using it.
Prototype, imo, shows its age, a good, solid library, but there are parts of it that are not needed and development on it is slow and there is no easy way to extend it with plugins. Cannot be used with mootools.
Mootools is a real javascript framework, providing excellent OO features, including classes and a solid inheritance model. If you want to use inheritance in your javascript, you should go with mootools. I often find this to be overkill for javascript, however, so I go with jQuery. Cannot be used with Prototype.
And finally, Dojo. Another excellent library, but somewhat bloated and has poor documentation, has a build process to cut down on the bloat. Also, some key features missing from its core library while the core includes features that belong in plugins. Can be used along side other libraries.
YUI, I only have experience with their 2.0 version and found that I hated it because I didn't want to type YAHOO.path.to.my.widget 13,000 times or always have to save a long path in a variable to refer to. That chased me away quickly.
GWT - only use if you are a java developer that doesn't want to develop in javascript.
DWR - some server side code required, primarily an ajax library, but it does provide some cool ajax features that other libraries do not... like Comet.
jQuery is my library of choice because it is the best at DOM manipulation and events, and excellent plugin ability. Also has excellent documentation and is the most popular library. Can be used side by side with other libraries.
Sizzle - CSS selector engine. Just finds DOM nodes via css selector syntax and returns them. Used in jQuery, going to be added to Dojo and possibly Prototype. Mootools will not be using it.
Prototype, imo, shows its age, a good, solid library, but there are parts of it that are not needed and development on it is slow and there is no easy way to extend it with plugins. Cannot be used with mootools.
Mootools is a real javascript framework, providing excellent OO features, including classes and a solid inheritance model. If you want to use inheritance in your javascript, you should go with mootools. I often find this to be overkill for javascript, however, so I go with jQuery. Cannot be used with Prototype.
And finally, Dojo. Another excellent library, but somewhat bloated and has poor documentation, has a build process to cut down on the bloat. Also, some key features missing from its core library while the core includes features that belong in plugins. Can be used along side other libraries.
YUI, I only have experience with their 2.0 version and found that I hated it because I didn't want to type YAHOO.path.to.my.widget 13,000 times or always have to save a long path in a variable to refer to. That chased me away quickly.
GWT - only use if you are a java developer that doesn't want to develop in javascript.
DWR - some server side code required, primarily an ajax library, but it does provide some cool ajax features that other libraries do not... like Comet.
Glow - YAJL. Unnecessary.