If you want to run a mail server on your PC I would advise to download
XAMPP as it comes with a complete web and e-mail server solution and is quite stable (ran a website on it for 2 years).
Now for the problen that you need a domain. You must search a place where you can register a domain with a DNS service that will allow you to set the DNS to your PC. For a .com domain be prepared to pay $10. Other domains might be available for less or free.
Once you registered, go to the DNS changing and put in the DNS of your PC.
Warning: you will need your PC to be on 24/7 or mail that is sent while it is switched off will not arrive. When running your own domain and mailserver you are responsible for keeping it up and if the server is down whatever people send will be refused or vanish in to thin air.