When you register a domain name, you are actually paying for the right to use that domain name for a defined period of time, usually between 1 and 10 years. You will have to pay a renewal fee, after your initial registration period expires, to continue using the name or face losing control of the domain name.
You do not automatically have the right to keep the name forever, although in practice, as long as you continue paying the annual fees and the name does not infringe on a trademark or service mark, it is essentially yours until you decide that you no longer want it.