This DBMS gets the number one nod mainly because the community version is free and is a great platform to begin learning on. There are, of course, commercial versions of MySQL for sale once you get to the point where you are developing large scale commercial applications, but getting started will cost you a big fat nothing.
Initially MySQL was available on a small set of platforms, Linux for the most part. However, now you can put MySQL on pretty much anything that you can imagine including Linux, Mac and Windows.
There is also a community version of their MySQL Workbench tool which gives the beginner a nice GUI (graphical user interface) to visually design and work with database tables. MySQL Workbench is reasonably intuitive, helps the beginner learn syntax by allowing the user to see syntax created by the tool when a task is designed, and has decent help documentation. Better than the actual help documentation you can find a ton of online tutorials, blogs, articles, etc. on developing in MySQL.