http://www.icann.org accredits various registrars to handle domain registration
(see: http://www.icann.org/registrars/accredited-list.html ).
If you don't see your registrar on this list, it is because many registrars are actually resellers of one of these companies, and often offer a lower price per single domain, than the actual registrar itself (sometimes buying in bulk, and passing the savings on to you).
When you get a domain, in order for it to resolve (go somewhere when you type it into a browser), you need to enter nameservers (example: ns1.something.com, ns2.something.com). This nameserver contains the zone file for the domain.
The zone file points the domain name to the proper server (the server in which the domain is set up), and directs subdomains, email services, etc.
The domain must be set up on a server (you can't just point it to a server and hope everything will work, it won't). The server contains a file (httpd.conf) which has all the info such as the directory of the site's files, and other needed configuration lines.
That's the short version in a nutshell...