MySQL is a relational database management system (RDBMS) developed by Oracle that is based on structured query language (SQL). A database is a structured collection of data.
The database is free and open source with a commercial license available (MySQL is now owned by Oracle after they bought Sun). Common applications for MySQL include php and java based web applications that require a DB storage backend, e.g. Dokuwiki, Joomla, xwiki
MySQL has one major advantage, since it is free, it is usually available on shared hosting packages and can be easily set up in a Linux, Unix or Windows environment. If a web application requires more than database, requires load balancing or sharding