MySQL is one of the supported databases that underpins a Moodle installation.
MySQL describes itself as "the most popular Open Source SQL database management system, is developed, distributed, and supported by MySQL AB. MySQL AB is a commercial company, founded by the MySQL developers. It is a second generation Open Source company that unites Open Source values and methodology with a successful business model."
MySQL comes with an array of storage engines. The popular ones being MyISAM and InnoDB. Since MySQL 5.5.5, MyISAM was dropped as default and InnoDB was made the default storage engine of choice. InnoDB is more well supportable than MyISAM due to known issues with MyISAM. MyISAM and non-ACID master-master setups can have issues which sometimes prove difficult to support.