Some totally free databases:
MySQL had good documentation and a very active and public user community; however, it has more non-standard things about than Postgres.
Postgres is more "industrial strength"; the documenatition and user community are good.
Both Postgres and MySQL are full-fledged database server products; they need to be set up and administered and they run continuously. Neither of them require a huge amount of computer resources, but they do require some, especially memory.
HSQL and Derby (no relation, formerly IBM's Cloudscape) are embeddable Java-based databases; this means they can literally be part of your Java application, without the server setup. Both are newer and less-featureful than MySQL and Postgres. In particular, I seem to remember that one of them doesn't support transactions, which are essential for multi-threaded updates to a database.
There are other totally free good choices too, such as Firebird, but I can't really speak to their pros and cons (a guy can't know everything...)
Additionally, some commercial database vendors make their programs available for free for development/training purposes. In particular, Oracle allows developers to use their unrestricted DB for free, limited to one single user. They also produce a free "light" database with some built-in limits that is both multi-user and freely usable in commercial projects. You have to register, but it's a free download.
The ACM's sigmod maintains an incomplete list of database software that is free for either academic or personal use, here:
http://www.sigmod.org/databaseSoftware/ not all of them support JDBC (in fact, the majority don't).
[ April 18, 2006: Message edited by: stu derby ]
