MySQL

MySQL has been around for a long time and was originally developed as Open Source and by a not-for-profit company. However the MySQL company was acquired by Sun in 2008, who were themselves bought by Oracle in 2010, thus MySQL is now an Oracle product. It remains Open Source, however there are paid for/licensed versions.

Being Open Source and commercially owned does not suit everyone so there are a couple of forks, the most popular of which is Open Source Database, Enterprise Database | MariaDB