The 3 Regs - Registry, Registrar, Registrant

Registry: The organization that maintains the database for a given Top Level Domain (TLD: .com, .net, .org...). The database they maintain houses the domain name and information as to who registered it and how long the domain name has been registered for.
Registrar: A company that runs a website where you can register a domain name. There are many domain name registrars to choose from and most offer you the opportunity to register a number of different TLDs. They are your connection to the different registries.
Registrant: The person, company, or organization that a domain name is registered to.

Rules for Domain Names

A domain name has rules as to how many and what characters are allowed. Some can be as long as 64 characters plus the .tld. All domain names can be made up of only numbers, letters, and dashes. No domain name can start or end with a dash. There are other rules involved as well and each registry can be different. Before registering a domain name you can check to see what the rules are, but it's not necessary since the registrar you choose to register a domain name with will make sure you don't register something the registry does not allow.

The registries, registrars, and registrants all have rules to follow as well. These rules are administered by the ICANN (Internet Corporation for Assigned Names and Numbers) organization.