Domain registration is the process that a registrant (e.g., you) performs at a registrar (e.g., easyname) to register a new domain (e.g., easyname.com) under a certain top-level domain (in this case .com). Domain registrations are always chargeable. By registering a domain, the registrant becomes the owner of the domain, provided that he registers himself as the owner of the domain. During domain registration we make sure that you comply with all the formal rules set by the respective authority.
What those formal rules are, depends on the respective top-level domain. For example, some TLDs require the domain owner to be resident in a particular country. Other TLDs require that you provide some personal identification, such as a passport copy. What the requirements are, can be read on the detail page of the respective TLD. You can get there through the overview of all available top-level domains.