This is regarding the Moodle Messaging system.
The Messages feature allows you to communicate with other Moodle users. When the user you are messaging is online a pop-up with your message is displayed to him/her. When offline, depending on the user's settings, your message is sent as an e-mail notification or as a pop-up display the next time the user is online.
You can either reply directly from your e-mail (which moves the message outside of Moodle) or by logging in to Moodle to reply (which keeps the message inside Moodle).
If you reply to Moodle message via your email you must make sure that you sent a reply to the sender, not to the firstname.lastname@example.org, as it is set up by default.