A JavaScript library or Javascript Framework is a library of pre-written JavaScript controls which allow for easier development of JavaScript-based applications, especially for AJAX and other web-centric technologies.

There are many JavaScript and Ajax framework available in the web. Its not easy to choose which one is the best. Last month i research a lot about these frameworks. I think a lot of Web developers are also confused which one is good.
I am listing here some of the good frameworks:

1. jQuery –http://jquery.com/
   “jQuery is a fast, concise, JavaScript Library that simplifies how you traverse HTML documents, handle events, perform animations, and add Ajax interactions to your web pages. jQuery is designed to change the way that you write JavaScript.”
2. MooTools –http://mootools.net
   “MooTools is a compact, modular, Object-Oriented JavaScript framework designed for the intermediate to advanced JavaScript developer. It allows you to write powerful, flexible, and cross-browser code with its elegant, well documented, and coherent API.”
3. script.aculo.us –http://script.aculo.us/
   “Script.aculo.us provides you with easy-to-use, cross-browser user interface JavaScript libraries to make your web sites and web applications fly.”
4. Prototype –http://www.prototypejs.org/
   “Prototype is a JavaScript Framework that aims to ease development of dynamic web applications. Featuring a unique, easy-to-use toolkit for class-driven development and the nicest Ajax library around, Prototype is quickly becoming the codebase of choice for web application developers everywhere.”
5. Dojo Toolkit –http://dojotoolkit.org
   “Dojo is an Open Source DHTML toolkit written in JavaScript. Dojo aims to solve some long-standing historical problems with DHTML which prevented mass adoption of dynamic web application development.”
6. Yahoo! UI Library (YUI) –http://developer.yahoo.com/yui/
   The Yahoo! User Interface (YUI) Library is a set of utilities and controls, written in JavaScript, for building richly interactive web applications using techniques such as DOM scripting, DHTML and AJAX.
7. Ext JS –http://extjs.com
   “Ext JS is a cross-browser JavaScript library for building rich internet applications that includes high performance and customizable UI widgets, a well designed and extensible Component model, and an intuitive, easy to use API. “

There are a many options when choosing a JavaScript framework for your project.
My favorite is jQuery, as you can find lots of examples and plugin on internet for jQuery. One plus point of jQuery is that it is supported by large community.

cheers!!
Joginder poswal