Building on Tim's comments, which database you use depends on whether you need an entirely 'free' database or not. The following discussion details some options
https://groups.google.com/forum/#!msg/mitappinventortest/E0ggiuQEUlE/03zQUp1mCQAJ . Is the database for your personal use or a commercial activity?
An sql (standard query language) or relational database offers in-built ability to sort or categorize data; a flat file database requires the developer to build his/her own sorting, data manipulation routines with Blocks. Both types of data storage work well.
Any commercial database that has a public api can theoretically be used with App Inventor. Only CloudDB (a Redis open source database), TinyWebDB and FirebaseDB (deprecated by MIT. AI cannot access most of the Firebase capabilities) use a App Inventor controls to simplify access. There may be a third party extension that would allow easy use of other available databases. All functions of an api might not be available using App Inventor 2. Note that Firebase that AI can access is not a relational database.
If you are serious about building an Android app that access an on-line database for commercial purposes, consider coding using a professional compilee like Android Studio or B4A etc.These compilers have the libraries to manipulate most data bases. Consider either hosting the database on your own server or use a database that provides that service (see link above).
What database to use depends on your skill, app requirements and pocket book.