Messaging is a Joomla! native component allowing administrators (and more generally all users who have access to the administration and to this component) to communicate privately. This component is available in the administration part of the site only, therefore it cannot be used in Frontend.
You can access this messaging system by selecting Components (1) → Messaging (2).
You'll be taken to the following page:
This component is very easy to use:
1 - Search for the messages that were sent to you by other users.
2 - Send a new message.
3 - Customize the parameters, you can for example choose to be notified by email when you receive a new private message.
How to send a private message?
To send a new private message, select Components → Messaging, then click the New button:
All you have to do is to enter a recipient and a subject, then write your message and finally hit the Send button.