This is a very simple thing to do, and once you see it, I'm sure you'll get it.
Simply create your form containing the input box for the email and the submit button. Set the form action to process your PHP script, i.e. form action=http://yoursite.com/phpscript.php.
Inside your php script, set up your database connection information.
$user = "yourusername";
$pass = "yourpassword";
$db = "yourdatabase";
$link = mysql_pconnect("localhost",$user,$pass);
mysql_select_db($db);
$query = "INSERT INTO tablename (FieldName) VALUES ('$InputTextBoxName')";
mysql_query($query, $link);
mysql_close($link);
Hope this helps.