This is the issue when there are all data filled out in your affiliate signup form, everything is correct and clicking the Signup button does nothing. There is no sign of error, but if you check it with Firebug, you will see an error there:
uncaught exception: com.google.gwt.event.shared.UmbrellaException: One or more exceptions caught, see full set in UmbrellaException#getCauses
The reason why you see this error is probably because of your recaptcha settings. Please check your recaptcha settigns in Configuration> Affiliate signup. The problem is probably caused by missing recaptcha private and public keys.