MySQL is the most popular free open-source (GNU GPL)  relational database management system (RDMS). It is used by many popular web applications (wordpress blog, joomla cms, drupal, phpbb forum, galleries and many others) for data storage of settings, text and even binary data (photos or other files). MySQL is often mentioned with together with PHP web programming language, and with Apache HTTPD and Linux OS they are part of the popular LAMP setup. Lots of tools have been developed to ease the work with MySQL databases. The most popular is phpmyadmin tool (written in PHP) that allows you to easily manage databases, create/edit tables, execute queries, browse databases and other tasks.

DreamHost MySQL

Dreamhost supports MySQL on their servers. Interesting option is that you can have your own VPS dedicated MySQL queries in case your site or service are very popular. With dreamhost control panel you can easily add new databases, setup mysql hostnames, users etc. You can have unlimited databases.