In order to have a public website, you must have a registered domain name. You can purchase a domain name through a plethora of domain name registrar services such as Networksolutions, eNom and GoDaddy.
Once you have a registered domain name, you must have a web hosting service. The web hosting service is where you upload the files used in your website. Your web hosting service also includes many other items and services that vary widely. The most basic services that you need to have:
Once you have a web host, you need to register your Name Servers from your web hosting service with your domain name registrar service. This is how the World Wide Web works to enable you as web page viewer to type in an URL or click on a link and actually go to the website that you typed or was included in the link you clicked on.