SQL databases implement the powerful relational database paradigm, a way of thinking about data that is based in the mathematics of set theory. This makes them (in theory) more powerful and safer than other kinds of databases that preceded them. While relational databases started life on mainframes, MySQL is an example of how the concept can be adapted to desktops. After all, your desktop is probably a more powerful computer than the 'big iron' mainframe of the 1960's!
MySQL started as the project of an independent company, which was subsequently acquired by Oracle. A free and open version is still available, called the "community" server.
The MySQL download for the Community Server is at http://www.mysql.com/downloads/mysql/. If you just got http://www.mysql.com/downloads/, you'll be offered a Windows installer that includes a whole bunch of additional software and connectors, over 200 MB of stuff! We don't need all of that to test out WordPress, which is the objective of this exercise. Just grab the installer for the Community Server.