SQLite is an excellent little SQL compatible database that is embedded into the core Android system. This allows you to leverage a complete SQL database without having to ship one with your application (which will raise your code size dramatically). This makes it the most common tool for storing structured data on Android, but it's by no means the only option.
For many applications that require real-time synchronization with a server, people use Firebase Database. Firebase is a Google cloud product that includes a powerful document database that synchronizes its data in real time, all the way to the client. This means that an event is triggered on a client when any of its data is modified from outside, making it suitable for chat and messaging applications. However, tools such as Firebase require a large additional client-side API, tie your application to a service, and are very difficult to port your application away for later. Applications built with them may also...