I presume you already know how to show different pages using Movie Clips. Just make sure your "contact" Movie Clip contains input text fields with certain instance names, and then make a button that, once pressed, takes the content of these text fields and sends them to a server-side script. Your script will return something, and based on that information you can tell the user whether sending the contact e-mail failed or succeeded. It's good practice to empty the form fields after successfully sending.

That's basically what you need to do.

If you didn't understand every term I just used, you probably should start following ActionScript and, say, PHP tutorials online.

You don't need to have the file in the same folder as the SWF file. In your Flash file, you will use ActionScript to refer to the PHP file (which can be just "script.php", but also "folder/script.php").