mysql is a client/server database system. On the server (which CAN be your own computer, or it might be somewhere else) a daemon manages the database. On the client (which CAN be your computer, or perhaps a Web server running PHP) a client program or client library accepts SQL commands, communicates with the server, and does something useful with the results (generally displaying them).
If you are using a Web hosting company that supports mysql, then you probably will be using PHP with mysql client libraries, interacting with mysqld (the server daemon) running on the same machine. You don't need to download any binaries and install anything unless you want to play around with mysql on your desktop machine.
Most Web hosting companies provide a Web-based "control panel" tool such as phpMyAdmin for creating and manipulating database tables. A Web hosting company probably will limit you to a single database.
Try doing a Google search for "mysql tutorial" for some introductory lessons in using a relational database manager.