MySQL uses a "dual licensing" business model. Under this model, users may choose to use MySQL products under the free software/open source GNU General Public License (commonly known as the "GPL") or under a commercial license.
With the GPL license, MySQL is available free of charge. Users may download the software for free and modify, integrate and distribute it. However, GPL users must abide by the rules of the GPL, which stipulate that if a MySQL-based application is redistributed, the complete source code for this application must also be open and available for redistribution.
MySQL AB offers a commercial license for organizations that do not want to release the source code for their application. Commercial license customers get a commercially supported product with a level of assurance from MySQL AB, without the requirement that their MySQL-based software must be "open sourced". MySQL is able to provide a commercial license because it has full ownership of the MySQL code.
Proprietary applications (aka applications that do not share their source code and the rights to modify it with others) should always use the proprietarily licensed version of MySQL.
Software lisensed under a Free Software/Open Source license (such as the GPL, the BSD license, the Apache license and so on) can use MySQL for free.