OK I'll take a shot at trying to explain
abc123.com is a domain name registered to you
DNS information for the domain is held and controlled by the registrar where you registered the domain
In your DNS information there would be A records pointing to your IP address
abc123.com. A 192.168.0.3
mail.abc123.com. A 192.168.0.3
192.168.0.3 is only an example
Also in your DNS information is an MX record which would look like
MX 10 mail.abc123.com.
Then you would have aliases or cnames which would look like
pop3 CNAME abc123.com.
smtp CNAME abc123.com.
www CNAME abc123.com.
So to put all this together
If someone types "www.abc123.com" in their web browser your DNS information would be read and directed to your IP
If someone sends mail to "firstname.lastname@example.org" your MX record would be read and again the mail would be directed to your IP
Same applies for "pop3.abc123.com" and "smtp.abc123.com"
In the above scenario all you would have to do is make your mailserver responsible for mail sent to "abc123.com"
This is just a simple explanation and there are a few gotcha's for instance
For the above to work you must have a static IP address
Hope this helps and most of all makes sense