You typically need to do two things. One is to get the DNS set up and the other is to set up your sebserver.
To set up the DNS, contact the people that run your DNS. That may be register.com, or your hosting company or someone. Tell them that you want another A record added for your new subdomain. It should probably point to the same IP address as your current site.
Then, you can contact your hosting company and tell them that you want your webserver set up to serve content for the new name. They will probably make a new directory for you to put your HTML pages in.