As coldwerturkey suggested, this is a job for Javascript - not PHP. PHP doesn't know what a checkbox is, let alone how to do something when one is clicked. Javascript is a client-side scripting language that can accomplish what you're after. Simple example:
Show the form?<br>
<input type="radio" name="show_form" value="yes" onclick="document.myform.style.visibility='visible'">Yes<br>
<input type="radio" name="show_form" value="no" onclick="document.myform.style.visibility='hidden'" checked="checked">No<br>
<br>
<form name="myform" style="visibility: hidden">
Name: <input type="text" name="name"><br>
etc.
</form>
Obviously the above HTML code doesn't validate, but it should give you the general idea of how to go about doing this.
