You are essentially correct.
MySQL is a database server, which you can control via SQL commands.
It doesn't necessarily have to be located on a special "server-computer". In fact, a lot of developers have MySQL servers set up on their workstations for testing purposes.
And when I say "server", I mean a piece of software used by other programs, not a physical computer. (Keep in mind that hardware is just hardware. It is the software that defines it's purpose.)
MySQL itself does not have an interface. It is purely a "behind-the-scenes" kind of software.
You use front-end software, like the command-line tool that usually comes with MySQL, or even some sort of a GUI tool, to control the MySQL server.
The MySQL server itself accepts SQL commands. You use them to create and edit databases and their tables, as well as adding, manipulating and viewing the data contained in the tables.
(Check out
this article to see how that works.)
Many programming languages can interact with MySQL servers.
It is very popular to use MySQL as the back-end database for PHP driven websites. When that is the case, the PHP website can be considered a front-end to the MySQL database.
There are even web-based applications specifically designed to serve as front-end GUI applications to your MySQL database, like phpMyAdmin for example.
This is very different from programs like Access, where the GUI tool that made the database is usually used to view and manipulate the data as well.
MySQL is meant to be used by other programs as a silent background service, unknown to those who don't know it is there.