The Domain Name System is a system of mapping names to IP addresses. Because domain names are alphabetic, they're easier for humans to remember. The Internet, however, is really based on IP addresses. Every time you use a domain name, DNS translates the name into the corresponding IP address. It is similar to a phonebook for the Internet.
When a domain is registered the registrar contacts the top level root servers to indicate the domains DNS settings/zone file for the newly registered domain. This DNS information is then propagated downward to other DNS servers, eventually the visitors ISP's DNS servers are updated and provide the domain and IP resolution back to the client.