An infrastructure where the server tracks devices so it can send a message to the device. This contrasts with pull where the device is responsible for checking periodically for new messages.
On iOS and Mac apps, the prevalent push service is Apple's Push Notification Service which integrates with paid developer programs as well as enterprise customers that use it for mdm uses. The idea of a push service is also available on OS X server to let changes to contacts, calendars and email be pushed directly from a server to devices.
The system involves pushes over the network as well as local push notifications which are especially useful on a sleeping device or on iOS where programs generally do not run in the background and can thus set a reminder and have the OS deliver that notification at the proper time while saving CPU cycles and letting the app remain paused until it needs to run again. Wikipedia has a nice article on the evolution of APNS and links to other push protocls and services like Google's and Microsoft's implementation of this idea.