ICANN (Internet Corporation for Assigned Names and Numbers) is the international agency responsible for regulation. Visit the InterNIC Accredited Registrar Directory for links to sites to check.
Remember, a domain name is a convenience so that your visitors don't need to remember IP numbers. Your web hosting provider may offer domain name registration for free or at a reduced cost as part of your plan or you can use a separate registrar (often preferable if you plan to change web hosting companies frequently).
You can "rent" as many as are available or you feel you need. Check with your web hosting documentation. Often your plan will allow "pointing" or "mirroring" a number of domain names to your physical site.