In order to call javascript methods from a Java applet in a browser you need to use the Netscape LiveConnect API, there are some examples here.
Basically this is an API that is implemented natively by the browser and allowed a java applet to access the javascript engine of the browser.
Once you have that sorted then you need to call methods in applet from the desktop application, this is a little more tricky. The most easy way would probably be to have an Enum
that you exchange serialised instances of to describe the type of event.
There is an example of using sockets for communication here.