A very good RDBMS (Relational Database Management System) that is free and works well. I have used MySQL for many things over many years and it always performed like a champ. There are a large number of tools and APIs and the documentation is very good.
I have not had reason to use PostgreSQL much, but the times I did it worked well. This is also a true RDBMS.
Not an RDBMS, but a database in a more generic sense. Very fast and excellent when developing an application and arbitrary cached data is needed. Oracle bought sleepycat in 2006 and has, unfortunately, not kept the original link (http://www.sleepycat.com now directs to a 404 error). I will update this page once I find where in Oracle it has been digested to.
Other databases I know about but have not had a chance to examine:
Links to related pages:
Items marked with (nyt) indicate I have not yet tested it. I've heard
it's good (so it has made it on the list) but have not yet tested it
myself.