How can one close an Applet window or browser which contains that Applet when pressing an "Exit" button in an Applet?

Security permissions don't let an applet call System.exit(). Even if you signed the code and the browser/user trustd you, the browser wouldn't let you shutdown/close the Browser.

If all you want to do is close a Window/Frame created from the applet, this is no different than closing a window outside of an applet. Add a WindowListener.