Question

In other words, how can I tell if the person using my web application is on the server it resides on? If I remember correctly, PHPMyAdmin does something like this for security reasons.

Answer 1

You can use $_SERVER['REMOTE_ADDR'], which contains the
IP address of the client requesting it, as given by the web server.

$whitelist = array(
    '127.0.0.1',
    '::1'
);

if(!in_array($_SERVER['REMOTE_ADDR'], $whitelist)){
    // not valid
}

^{Note: the original version of this answer suggested verifying the destination hostname using $_SERVER[‘HTTP_HOST’], which is unsafe because it could in some cases be spoofed by the requester.}