You may have heard of AJAX. Simply put AJAX allows you to call PHP scripts (or other server side scripts) behind the scenes without making the user change page. AJAX will allow you to do what said, the process would be something like:
- On button click: grab the email in the textbox, fire off an AJAX request (behind the scenes) to a PHP script including the email address
- The PHP script sends the email to yourself saying that xyz has registered their email
Let me know if you want any help with the coding.