That are several Problems at once.
I would suggest learning some more about Java Swing, Frames/Windows and the AWT-Event-System
http://java.sun.com/docs/books/tutorial/uiswing/events/windowlistener.html
and then trying to use the JDIC tray functionality with a JFrame containing a “Hello World” JLabel.
After you have the tray working with all your needed features (close,minimize,maximize,tray context menu, etc.) you can exchange the JLabel with an JEditorPane to display a simple webpage and implement the state handling (e.g. remembering the page, etc.)
If you have this working, you can try to integrate the JDIC webbrowser component:
https://jdic.dev.java.net/nonav/documentation/javadoc/jdic/org/jdesktop/jdic/browser/WebBrowser.html
Since I have no knowledge about the state of the JDIC component, it might give better results to integrate the xhtml browser “flying saucer” instead:
https://xhtmlrenderer.dev.java.net/
If you are still not satisfied with the html rendering, you can embed mozilla via JRex:
http://jrex.mozdev.org/
To embed the Internet Explorer, you have to use the OLE/COM component under Windows. There is a software available (e.g. NewJawin) to allow this:
http://shareware.pcmag.com/product.php?id=60701&SiteID=pcmag
I strongly suggest to follow these steps in this order and stop at integrating “flying saucer”
Everything else might take a very long time to get working…