Citadel is installed as the only email server, connected to the Internet with a static IP address (no residential connections with DHCP or PPPoE).
Next, check that you can actually reach your Citadel server from other locations and there are no firewalls blocking the connection.
Verify that Citadel is able to connect to other mail servers (again: firewalls, routing, DNS) andd that it can resolve host names and IP addresses.
Then you need to check that your DNS is visible from the outside world that your names meet conventions. Also check to make sure that your address is not on any DNS blacklists
In this case your server is well behind an outbound-only Internet connection, such as a residential cable or dialup service. Your Citadel server is not directly reachable from the outside world.
You will need to poll your external Account via POP3 or IMAP.
Mail to external users will leave your Citadel via a smart host. Note that you can only specify one user on the remote server here.