Remy Sharp discusses various aspects of JQuery in this presentation from QCon London, 2008. The presentation begins with a short introduction of the main reasons for using a javascript library at all: Easier DOM Scripting, Cross-browser applicability, and makes programming less work, more fun.
Before discussing examples and 'how-to' issues, Remy suggests several reasons for using JQuery as a preferred library; including the fact that it is small, it plays well with other libraries, it has a large support community, and it is being used by some major players like Google and the BBC.
Most of the presentation deals with the specifics of the library: Selectors and Chaining, DOM manipulation, Events, Ajax, and simple effects. Details and examples flesh out the discussion. WatchJQuery: A javascript DOM Library to learn more about the JQuery library.