A domain is simply a web site address. The domain name system is just a way for the Internet to allow users to take advantage of easy to remember domain names such as www.this.com and have the system convert that to the actual numeric IP address. Users can therefore find websites, and give out e-mail and web addresses to your site without having to know it’s actual IP address or even how it is derived from the name they do know.
Domains are usually controlled and registered by central bodies. Buying a domain usually just means registering your ownership with that body for a limited period of time subject to renewal and following their rules in the mean time.