Like jQuery and YUI, MooTools offers nice, browser-independent shorthand for manipulating arrays, divs, spans, and whatnot. My favorite part continues to be the custom library construction tool that lets you select the functions you want. Check some boxes and get an entirely optimized version of MooTools with just the functions you need and none of the bloat you don't. That's lightweight.
A number of other libraries offer newer features -- for animation or data visualization or server-side processing or other niches -- or different ways of thinking about life in the browser. To get a close look at some of these newer options, I unpacked a number of libraries, wrote a few lines of code, instantiated a few objects, and pushed some code through a few browsers.
Animation and HTML5 game engines
One of the stated goals of HTML5, at least for some groups, is to replace the Flash plug-in, the gold standard for making sprites and letters dance across the screen. This change is slowly coming as the game industry and the presentation industry start to duplicate some of the sophisticated tools available in the Flash universe.
Mashi is an impressive example of how the sprites can be set in motion. It offers more than several dozen standard easing functions for moving sprites along a timeline.