It is possible to request access to the MySQL server that runs on the same machine as the webserver. It is intended for use by web applications though, not for heavy SQL usage.
You can request a database (and the corresponding account) by e-mailing email@example.com.
It is possible to access the database using various methods. Most commonly PHP is used. On the webserver itself it is possible to use the mysql command.
Manuals for MySQL can be found online. Please note that the online manual will probably describe a newer version than that BCF currently provides. Currently our server is running version 5.0.
In case the main site is down, we have a local mirror of the documentation.