There are two parts to the Ext framework. Ext Core was invented as a jQuery competitor but honestly if you're trying to do what jQuery does, just use jQuery. It's more popular, it'll be easier to find answers, etc.
ExtJS is a full-blown RIA framework with a rich library of widgets. You might say that it competes with jQuery UI but jQuery UI is a much less rich set of components so they're not really competitors as such.
Also, ExtJS is released under either the GPL or a commercial license. You may find that to be a problem.
For rich Javascript frameworks, I tend to favour YUI or possibly SmartClient.