Several computers linked to the Internet host part of the DNS database & the software which allows others to access it. These all computers are known DNS servers. No DNS server has the whole database; they have a subset of it only. If a DNS server does not have the domain name requested through another computer, DNS server re-directs the requesting computer to another DNS server.
The Domain Name Service is structured like a hierarchy alike to the IP routing hierarchy. The computer requesting name resolution will be re-directed 'up' the hierarchy till a DNS server is found which can resolve the domain name in the request. Top of the tree are the domain roots. Some of older, more common domains are seen close to the top. What is not illustrated is the multitude of DNS servers around the world that form the rest of the hierarchy.
While an Internet connection is setup (for example for a LAN or Dial-Up Networking in Windows), usually one primary & one or more secondary DNS servers are denoted as part of the installation. This way, any Internet applications which need domain name resolution will be capable to correctly function. For instance, when you enter a web address in your web browser, first the browser connects to your primary DNS server. After getting the IP address for the domain name you entered, then the browser connects to the target computer and requests the web page you wished.