A MySQL process or MySQL server (or even MySQL server process) refers to the actual running MySQL software.
A MySQL database, on the other hand, is a collection of related tables within a MySQL process that share a common purpose, like running a forum or blog application.
A MySQL process may contain more than one MySQL database. (If it doesn't have at least one, it isn't very useful.)
MySQL processes are identified by their DSN or hostname, which is of the form example.db. You chose this when you created the MySQL process, and you can see it on your MySQL panel in the user interface.
MySQL databases are identified by their database name. You can create these from phpMyAdmin.
Sometimes the term MySQL database gets misused to mean the MySQL process. This usually happens when the process only has one database. We try to avoid that usage, because it is ambiguous and confusing, and we encourage you to do the same. But it happens, so if someone says "MySQL database" and you aren't 100% sure which one they mean, ask them. "Do you mean the MySQL process itself or the database within the process?" The confusion you avoid may be your own, because you have to know which name goes where when you configure your application.