Message Notification is a feature that allows you to turn on, or off, announcements that you have messages waiting. Normally, when you are online and another online player sends you a message, you are immediately notified (once you have finished whatever menu function you are in the middle of.) With Message Notification turned off, you get no announcements that you have messages waiting, but must use the Check Messages function to see if any messages are waiting. A person sending messages to someone with message notification off will see a "notify off" warning after they have sent a message.
This feature is no longer used now that the game and message systems are separate.