Currently, Apple does not allow 3rd party iPhone applications to run as background processes. Instead, 3rd party iPhone apps must shut down completely after their use. Push notifications was Apple's solution to allow applications to receive notices while they are not the active application.
myPhoneDesktop takes advantage of this feature to alert you when new message has been received.
Tapping View button will launch myPhoneDesktop app and depending on how you configure application rules, myPhoneDesktop will process received data (i.e. copy received text to clipboard, open received URL in mobile Safari, initiate call to received phone number, save received image into your Photos library, etc).