The first thing to figure out when choosing a JavaScript framework is
identifying your need correctly:

• Need 1: DOM Manipulation (Enhancing your site by adding behavior) : If you’re looking only to enhance your site by adding behavior and to extend it with plug-ins of your own you can do it with jQuery. I.E. if DOM manipulation is the key feature you need, then JQuery is the best. Ext JS has recently released a free library Ext Core to counter JQuery and it seems to be pretty good.
  

• Need 2: Making widgets, custom JavaScript based GUI : Dojo is good bet here. YUI also provides good widgets.But if you want to give a more desktop like look and feel (windows, panels), EXT JS is the best. Though important to note is that its paid (developer license when you go to production).
  

• Need 3: Building a large application that will do a lot :Dojo is fantastic for building a large scale application that will do a lot, and end up with a ton of JavaScript. Everything you need will be found there.
  

• Need 4: Creating rich dashboards and admin consoles :Ext JS has a great usage in creating here. Its already created look and feel makes it very easy to reuse. But again its paid. It provides a lot of extensible, uniform-looking components (layout manager, templating support, trees, grids, tabbed panels, windows, etc.).
  

Some Criteria that you need to take into account:

• Criteria 1: Learning curve and speed of development : JQuery wins here, as its easy to learn and uses less lines of code. But its basically for DOM manipulation. Dojo is huge but then you can learn only what you want to use. Prototype fits somewhere in between these worlds. It is small enough to feel small (not a huge library to learn) yet large enough that you don’t jump out into creating a lot of your own code.
  

• Criteria 2: Code readability and Maintainability : Prototype is much more readable than JQuery. Ext Core and Dojo also are pretty readable. Complex JQuery code is difficult to maintain and to reuse.
  

• Criteria 3: Speed and Code Size : For basic JavaScript functionality, you could even use Robert Nymans DOMAssistant or DLite. ExtJs (not Ext Core) is comparatively heavier than the rest.
  

• Criteria 4: Good Ajax support :Generally all frameworks now a days have Ajax support but JQuery seems slightly lacking here.
  

• Criteria 5: Coding Style : JQuery doesnot follw what we could call the traditional class based style of coding but is more intutive. This is why its great for designers wanting to add some dynamic magic to their webpage. But a traditional developer will find better coding style followed in Prototye, Ext Core, Mootools and even Dojo. This makes the code more maintanable.
  

• Criteria 6: Cost : Well this is only a problem with ExtJs, but i bet they have got things in their library that are worth the money. Note: Ext Core is not paid though.

Some Good Resources which helped me compile the article:

Find the questions to ask when choosing a framework here:
http://blog.reindel.com/2007/10/30/how-to-choose-a-javascript-framework/

Find an overview of JavaScript libraries here:
http://www.slideshare.net/jeresig/javascript-library-overview-presentation

Great resource to work with JQuery:
JQuery : http://visualjquery.com/1.1.2.html

List of Pros and cons of JQuery, ExtJs and Dojo
http://fictionalrealm.com/2009/04/javascript-ui-framework-comparison-or-why-i-choose-extjs-jquery/

Ext JS vs JQuery
(http://coderseye.com/2008/why-i-didnt-switch-from-jquery-to-extjs-after-all.html)
(http://forum.lessthandot.com/viewtopic.php?f=44&t=2660)

Prototype vs Jquery
http://blog.thinkrelevance.com/2009/1/12/why-i-still-prefer-prototype-to-jquery

JQuery vs Mootools
http://jqueryvsmootools.com/

JQuery vs Dojo
http://almaer.com/blog/why-i-often-prefer-prototype-too

Performance comparison between mootools, jquery, prototype, yui and dojo
http://blog.creonfx.com/javascript/mootools-vs-jquery-vs-prototype-vs-yui-vs-dojo-comparison-revised:

Some Positives and Negatives about JQuery: http://stackoverflow.com/questions/176324/why-does-everyone-like-jquery-more-than-prototypescriptaclous-or-mootools-or-wh