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When the class Applet was defined, it included a method called destroy(). When an Applet's containing window is closed (and other times) this destroy() method is called. When you want to use the...

Ok I read your edited post. It is better not to edit, as when I (we?) read, very rarely do I reread previous posts, just the new stuff...
So some code...

public class MyApplet extends Applet {
...

Well it is a method of applet, not Thread. Second it is a method of applet, and not your class, so an Override notation would be useful. Follow the link in post #8 for more information on it

You may have been writing at the time I posted. Just so the thread gets a bump.. See post #8.

destroy()
Override this method and drop your resources within there.

Well I am just guessing you are using jframe. Seeing code would simplify the guessing what you are using process.
Have you investigated the windowClosing method? This method is called when the ...

With what little you provided, maybe call jFrame.EXIT_ON_CLOSE or use the windowClosing method of a WindowListener. If that seems foreign to you, post more code so we can see what you are working with