Is there an official organization for domain registration?
I've seen many sites that provide this service.
Name.com, Go Daddy and Register.com are domain name registrars, of which there are many and all of whom have the authority to register a domain name on your behalf.
The Internet Corporation for Assigned Names and Numbers (ICANN) is the authority from which the registrars get their ability. ICANN helps maintain the top level domains like .com, .net, .uk, etc.