Once you have validated the form and stored the results you would call mail()

Assuming your have already taken the users $_POST['email'] variable, validated it and stored it as $email further up the script then ....

<?php
$to = $email;
$subject = 'Form submission at www.yoursite.com';
$message = 'This is an auto generated response from www.yoursite.com\r\n\r\nThank you for your submission.\r\n';
$headers = 'From: webmaster@yoursite.com' . "\r\n" .
    'Reply-To: webmaster@yoursite.com';

mail($to, $subject, $message, $headers);
?>