Ok, download CakePHP 1.2 and create a basic website with some scaffolding or something. Include Prototype or jQuery and think about a part of your page you want to replace with the result of an Ajax request.
Suppose you have a scaffolded list of customers. When you press the View link, Cake redirects you to customers/view/123. You can catch the link and call some Javascript to load the result of customers/view/123 in a div on your index page. Quite easy, but you now see your header twice. Not cool. Hmmm....