MySQL is the world's most popular DBMS, and is open-source, free, and very, very fast. It owes a great deal of its popularity to PHP, as when PHP 3 started to take off, the PHP developers worked very closely with the MySQL developers to make sure the two integrated very easily on the web.
MySQL achieves its incredible performance by not providing much of the enterprise-level functionality seen in other databases, but it is now working to add them in without sacrificing too much performance. MySQL 5.1 is the latest stable version of the DBMS at the time of writing, and plans are already in place for 6.0, which is destined to add more functionality.
MySQL runs on many platforms, and is open-sourced (using the GPL licence) so you can look at, edit, and redistribute the source code if you so wish. MySQL allows you to select from a number of "table handlers" (aka "storage engines") - each table can be of a different type, with MyISAM being the default. MyISAM tables are very fast for reading, but slow for writing. Another popular option is the InnoDB table handler, which are much slower, but have full support for transactions and foreign keys. If you use MyISAM, MySQL will not support transactions at all.