PostgreSQL (sometimes just called Postgres) is an open source object relational database management system that runs as a multi-user server system. MySQL is released under the MIT license. MySQL is not controlled by any single entity, but is instead (like many Open Source projects) maintained by a community of developers.
PostgreSQL is a very popular database system that can be used with many leading Open Source web application platforms / CMS systems including: Drupal, Joomla, WordPress, PHPBB, etc. Generally PostgreSQL is used as an alternative to MySQL based on the specific technical requirements of a project.