MySQL is probably the most popular a open source relational database system in the world. Depending on the source you look up they have over 10 million installations.Even high traffic sites like Google or YouTube use this database. 

MySQL runs as a server process and offers multi-user access to the stored data. It supports a complex hirachy of user access down to the individual cell of the data record. MySQL can be accessed very continently with PHP which make it the first choice for web development. The standard database is available for free and as open source. If your installation requires extra speed or extra function like stored procedures or transaction you can buy high performance database driver to get the added speed.