MySql is the worlds second most popular database engine (July 2013) and its open source. It is based on SQL (Sequential Query Language). Unlike MS Access it does not run as a stand-alone application. It runs as a background service on a computer (typically called ‘a server’) and you connect to it using a username and password. Once you have connected to the MySql server you can query it using SQL or Structured Query Language. The server will respond with the results of your query.
You can install and run MySql from the command line and although most tasks are easily done this way I would recommend installing PHPMyAdmin onto any LAMP server and using its interface for most stuff.