MySQL is a common database system used in GNU/Linux. It is extensively used as the backend database server for the Apache web server. To quote the Debian information about mysql-server:
The main goals of MySQL are speed, robustness and easy to use. MySQL was originally developed because Tcx needed a SQL server that could handle very big databases with magnitude higher speed than what any database vendor could offer. They have now been using MySQL since 1996 in a environment with more than 40 databases, 10,000 tables, of which more than 500 have more than 7 million rows. This is about 50G of mission critical data.
Under Debian install the package mysql-client for connecting to a mysql database server (possibly remote). To run a database server on your local machine you will need to install mysql-server. Install mysql-doc for the full documentation for MySQL.
The package mysql-navigator provides a very effective (but non-Gnome) GUI to MySQL. You can manage the MySQL database server using this. When started, open a connection to the MySQL server on the local host with user root. Most (but not all) admin operations can be performed through the navigator.
A Gnome-based interface, and somewhat easier to use interface, is provided by the gmyclient. This is not yet packaged for Debian but using the alien package creates a Debian package that installs just fine.
A beginner's tutorial on using MySQL is available from http://www.devshed.com/Server_Side/MySQL/Intro/. Also see the full user manual in /usr/share/doc/mysql-doc/index.html.