In fact, you can use every database which is accessible through a JDBC driver. Almost all self-respected RDBMS vendors provides a fullworthy JDBC driver for download at their homepage. Just Google "[vendorname] jdbc driver download" to find it. Here's an overview:
This way you can use the JDBC API transparently to access either of the databases.
As to which database to choose, just look at the features, robustness, performance, etc the RDBMS provides and the budget you have -if it isn't freeware. I myself tend to prefer PostgreSQL.
Instead of a fullfledged database server, you can also consider an embedded Javabased database, such as Sun Oracle JavaDB, Apache Derby, HSQLDB or SQLite, each which are of course accessible through the JDBC API the usual way.