MySQL
MySQL is an open-source Relational Database Management System, and is one of the most widely used RDBMS in the world. It is also a crucial component of the AMP (Apache, MySQL, PHP) stack.
Being an RDBMS, MySQL uses a version of the Structured Query Language (SQL) to create, modify, and delete data from a database. While MySQL is often paired with PHP, it can be and is used by other programming languages as well.
MariaDB
While MySQL is an open-source product, it has changed owned a few time over the years, and is currently be Oracle, the creator of the one of the most popular, proprietary RDBMSs in the world. Because of the obvious conflict of interest, there were concerns through the developer community that Oracle would convert the MySQL to be a proprietary software or stop supporting MySQL all together. This fear lead some individuals to take action.
In 2010, before Oracle had officially acquired MySQL, Monty Widenius, one of the original creators of MySQL, created MariaDB. MariaDB was and still remains a forked copy of MySQL, but with a fully open-source license. The team behind MariaDB continues to ensure that it remains compatible with nearly all of MySQL features, making it easy to switch between the two database management systems.