From MMA Tycoon Wiki
You can use the in-game mail system to send messages to other users, to arrange business ventures, contracts or just to keep in touch.
Sending a message
To send someone a message, go to their manager profile page and click the private message link in their contact information. This will bring up a pop-up box for you to send them a message.
Receiving a message
If you receive a message, it will display a "1" next to the "mail" option in the menu. The same system applies to fight offers, contract offers and sponsorships too, so you don't have to keep checking the pages manually.
Replying to messages
Once you have received a message from someone, you can reply to that message in the mail system, as you would reply to any email, in e.g. hotmail.