As already suggested, checking an e-mail box (POP3 or IMAP) is a valid way to process e-mail. However, if you're looking for realtime processing, it is indeed possible to have the mail delivered directly to php as standard input.
You basically tell the mail server to deliver the message to a local script.
You'll have to delve into CLI PHP, and configure your mail server (which will need to be on the same server as you processing script). So it's not an entirely simple process. But it is possible. I currently run a PHP powered e-mail discussion list using this method.
I configured my long ago, so I can't recall all the details; but this article should point you in the right direction.