Notifications enabled on app make the app run on background?

Let me start with an example, Delta's airline app called "FlyDelta" has an option in it settings to enable/disable "Get check-in reminders & status notifications". My question is, if I enable this "Get check-in reminders & status notifications" will the app from that point on run in the background at all times (even after restarting the phone or killing the app)? I'm a bit confused on how notifications work. I know that Google has a very nice infrastructure to push notifications to an app, but does the app has to be running in order to consume the data? any help explaining this specific app behavior will be greatly appreciated.