The "mysql" database contains vital information about the other databases, usernames and passwords in your MySQL process used when it starts up. It is a very important part of MySQL, and your process won't start if it's removed or corrupted. Unfortunately, MySQL will let you alter or remove it without any warning or special confirmation. Don't ever do that. Because people sometimes do, our system checks on this database periodically and if it detects a problem it will report your MySQL process as "Damaged" in our member interface.
MySQL 5.0 and later versions also contain a synthetic database called "information_schema" which can be used with SQL commands to obtain metadata about your other databases. Since this database is created entirely from information about other databases, it doesn't really exist, and attempts to alter or remove it will fail.
MySQL processes may also be created with a database called "test." This database serves no purpose and can be removed if you don't want to use it.