1) Your screen name does not conform to Javaranch rules. You have been asked before to change it - please do so.
2) An applet can't call System.exit due to security restrictions (unless it is signed).
3) What should happen if the applet is closed - should the web page it is embedded in be closed? That is tricky to do in the general case. Or does the applet open its own window? That should be easy to close, but the applet will still be there as part of the web page.
4) What error do you get? Show us the exception, and show us the relevant code section (not the whole applet).
Why do you want to have a button to close an applet? Typically the user will "close" an applet by going to a different web page, either by entering an address manually or clicking on a link. This is the way applets should be designed to work.