Use sql lite. Android supports it.
The sqlite database is just a file on an SDCard. It is not tied to the application, so uninstalling and / or updating the application does not affect the database. (if you did not write an update code to update your db :))
Sqlite is easy to use, compact and, obviously, can be easily stored on an SDCard. The ability to connect to it, etc. Runs in android.
You only need a web server database if you are going to share information from your application between devices / people. If data is only needed for your local device, then definitely use sqlite.
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