For lightweight, simple, no-frills, it's hard to beat DBD::SQLite. It does have its limitations though, one of which is that its implementation of SQL is a little less than complete and total. Nevertheless, it provides all of the essentials, so long as essential needs aren't too elaborate. You can read up on SQLite's implementation of SQL and its features at http://www.sqlite.com. SQLite's source is in the public domain, and the CPAN module is distributed under the same terms as Perl. SQLite, as others have mentioned, installs everything you need when you install the single CPAN module DBD::SQlite. And the databases are contained in single files.

If your needs are more involved, more demanding, and if you can live with increased complexity, then PostgreSQL is another very good option. Its implementation of SQL and its feature set are more complete, and include many advanced features. I doubt that you'll ever outgrow it, unless you become the next eBay or something. PostgreSQL is distributed under the same license as BSD, and is also free. You can find the database engine at http://www.postgresql.com, and the Perl drivers for it at DBD::Pg.

Both DBD::SQLite, and DBD::Pg are designed to be used with DBI.

Good luck!