Push notifications are sent to the iPhone as an alert which causes the phone to make a sound, vibrate, and leave a message on the "lock screen" of the phone.
In the popular app, "Words with Friends", for example, a push notification is sent to a user whenever the user's opponent has made a move and it is now the user's turn to make a move. The CNN app uses push notifications to send messages about "breaking news". These notifications are handy because, from the lock screen, the user is able to check their list of notifications and pick one they want to interact with. By selecting a notification, the app on the user's phone is automatically opened.
We've set our push notification system up primarily as a promotional tool for studios. Studios may, for example, send a push notification to announce a time-sensitive promotion like "20% off on all classes this week!". In practice, however, it's possible to use the system to send notifications for non-promotional events as well. If there is an upcoming class cancellation, for example, you might send a notification to announce the cancellation.
It's not uncommon for iPhone users, in general, to download an app, open it once after it's been downloaded, and then completely forget about it and never open it again. Push notifications are, in effect, a way to increase the visibility and interactive-ness of your app.