Sending A String from a Website to Arduino
On the yun, this functionality is built-in via the bridge. Here's an example.

Via the shield, you now end up writing your own webserver and responding appropriately. 

I'm guessing it makes more sense to use the client (the web browser) to make the calls than the backend php, but in either case, once the webserver is up and going on the arduino, you're just using it like any other server.