This is a rather generic question, so I am going to give you a lot of options.

One of the most common places to utilize the ClientAPI is in the development of a WebContol.  The ClientAPI Developing WebControls document walks you through the creation of the TextSuggest control.

Other areas within DNN also utilize the ClientAPI and are documented in ClientAPI MinMax and ClientAPI DragDrop.

A generic description on what the ClientAPI is is found in the document ClientAPI.  The callback capabilities are discussed in Client API Callback.

As stated in the ClientAPI document, the documentation for the api can be found at http://webcontrols.dotnetnuke.com/js/clientapitests/.  Take note of the links to the Unit Tests. 

A standalone app that utilizes the WebControls (which use the ClientAPI) can be found at http://webcontrols.dotnetnuke.com/.

Finally, you can visit the ClientAPI project page and read the latest blogs and look at the Downloads which contain the sample apps and modules utilizing the ClientAPI.

Note: If you experience problems clicking the links to the documents above, I suggest downloading the entire zip of all documents and looking at them locally.