The specifics of this kind of depend on the domain. Like if you get hosting service along with the domain, they will usually also offer a few email addresses to go with it that can be handled by the hosting company's systems.

Otherwise, the basics of it is, you need to have the mail server running on whatever system has the IP address pointed to by the domain. There are some complicated means to get around that, but you're on your own with those.

You then have your choice of storing all the email on the same system as your domain is being hosted on, or to have it just be forwarded along to another email address. Generally a pretty easy thing to configure either way. You don't even necessarily need to use the same server to send email from that domain. You can just configure your email client to say it came from that domain, and while if someone looks at the full email headers, may be able to tell otherwise, better than 99% of people will never know the difference.

If you want to know more, there are dozens of books on the subject. Some specific to a particular mail server package, such as Sendmail and Postfix, but there are also more generic network admin type books.