People's use of jQuery serves as good demonstration of its abilities. You may also want to try introductory examples, tutorials or plugins.

Webpage functionality demos:

• Stylesheet switching
• Sliding menu
• InnerFade

More complex and/or shinier demos:

• Image Gallery using jQuery, Interface & Reflections
• Chat
• Tetris
• jQuery puzzle
• Google map integration

Incidentally, note that you can use jQuery from the FireBug console on pages that use jQuery - might be interesting for a quick live test.

Quick example: Hiding a div

This box is a div that has class="hidetest". The jQuery $('.hidetest') selects this -- and all other elements that have 'hidetest' for a class (here using CSS. You can also use simple XPath).

You can then do something with it, for example hide it. The "Hide/show the box below" link above does both hiding and showing through toggling, using the jQuery $('.hidetest').toggle();