"Firebase Cloud Messaging (FCM) is a cross-platform messaging solution that lets you reliably deliver messages at no cost." That, I believe, is the best brief description of this service. It is actually part of a suite of many other services on the Firebase platform created and run by Google.

You have integrated In-app notifications, and now you shall see how to implement push notifications using FCM. 

In-app notifications basically means that the notification is triggered by and from within the app. Push notifications, on the other hand, are sent from an external source.