The name “MySQL” is sometimes used to refer to the MySQL database management system and sometimes to the company behind it. The company is more properly known as MySQL AB. MySQL is the DBMS product that MySQL AB owns, develops, and sells-that is, the database server software called “MySQL” and related products, such as client programs for communicating with the server and programming interfaces for developing new clients.
As a side note, the “AB” part of the company name is the acronym for the Swedish “aktiebolag,” or “stock company.” Thus, the name translates to “MySQL Inc.” In fact, MySQL Inc. and MySQL GmbH are examples of MySQL AB subsidiaries.