SQLite3 is an embedded database. The Android runtime has got this db engine baked into it, so you don't have to install it, its already there. Every Android application you create can use a SQLite3 db.
For old hands in Java, bad news, because you won't be able to use the JDBC API. There's a new set of API to learn if you want to do db programming in Android---the good news, the API is not very complicated. It won't take you an afternoon to learn.
Use case
Let the user type anything he wants on a free-form editable text component (EditText)
Get the runtime value of the EditText, then insert this record into the SQLite db
Read the whole table, display it in a scrollable, non-editable text display component (TextView)
Concepts
In web or desktop programming, we are used to creating the database using familiar tools like an IDE. In Androiddevelopment, you will need to master the SQLite APIs even as early as database creation.
The SQLiteHelper (found in android.database.sqlite.SQLiteHelper) holds the key to activities such as database creation and upgrade. You will also use this class to get a handle on either a readable or writable database. Think of it a replacement to the Connection object that you use to love in JDBC.
Next, learn to use SQLiteDatabase object. If SQLiteOpenHelper is the equivalent of the Connection object, SQLiteDatabase is the equivalent of the Statement object (in JDBC). If you need to perform an insert or a raw execSQL, SQLiteDatabase is responsible for that.
Finally, you need to be acquainted with the Cursor object. Think of it as the equivalent of the ResultSet object in JDBC. If you need a handle in actual rows and columns of your db tables, this is the object you will use.