The mail screen is where you will receive fight contracts, contract expirations (for fighters), offer confirmation or rejection from fighters/coaches, warnings for low funds and more.
Be careful not to sim past fight contracts! New fights and offers are generated after an event has completed.
If you do not accept a fight contract before the expiration date (1 week) it is removed and counted as a fight denial. 3 fight denials and your fighter is out of their respective organization.
They will get an offer to rejoin, but at the cost of their ranking.