We're all familiar with the names: YUI, Prototype, Dojo, JQuery, MochiKit, Tibco, Backbase, and many more. Some are free; some aren't. Some are well documented; many more aren't documented at all. Some support all the browsers you care about; some don't. Some are accessible and work well with assistive technologies; many don't. Some are spry and fast; some are heavy and slow. What all the prominent JavaScript libraries have in common is that they promise to save you time in building rich internet applications by offloading some of the heavy lifting that characterizes development in the browser.
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