$user is an instance of a class.
connection is a method in that class.
$mail & $password are parameters to that method.
This has nothing todo with arrays. what you mean would be:
$foo = array("key" => "value");
maybe this can help you:
http://www.webstockbox.com/php/7-tutorials-on-how-to-create-a-php-login-system/
solved To understand a line of PHP code about a connection