Firebase is a cloud based mobile platform owned by the giant search engine company Google ,which is designed to offer developers an easy to use SDK for quickly building back-ends for their mobile apps (or even web apps) without writing server side code .
It provides also many great tools for analytics and mobile app monetization .
Firebase includes many useful services such as
Firebase realtime database
Firebase Push service .
Cloud Messaging .
Firebase is a BAAS or Backend As a Service provider created in 2011 .
Firebase can be mostly useful if you are building a mobile app where you want synchronize data between all connected clients in realtime .
Firebase offers a realtime NoSQL database where you can store JSON objects .
Ionic Cloud is a set of cloud based tools for working with Ionic apps .You can use Ionic cloud to build and test Ionic based mobile apps using Ionic view and Ionic creator .
You can also use other services of Ionic cloud such as the Auth system which allows you to integrate normal email and password or social based authentication into your apps without writing custom server side backends.
Also recently Ionic has added another service similar to Firebase realtime database .The service is IonicDB which is still in beta (In the time of writing this post ) ,IonicDB is a realtime NoSQL database that allows you to have full fledged backends for your Ionic mobile apps without writing a single server side code .
Unlike Firebase ,IonicDB or Ionic cloud services are tightly integrated with the open source hybrid framework Ionic .
Both Firebase and Ionic cloud provides push notifications ,auth services and realtime NoSQL databases .