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Domain Name Server
I'll get to what a domain name server is in a second but here is the reason why they exist. Computers don't understand human readable domain names like BestBuy.com. Computers only understand numbers. So the domain name servers are responsible for translating that domain name, BestBuy.com into a number that it knows how to work with. This is known as an IP (Internet Protocol) address in geek speak. In essence, it is a directory of network address mappings to domain names.

Going back to our analogy, you can think of the domain name server as the Postal service. When you move to a new house, you give the postal service your new address and they associate your name with that address. It's the same with your website. When you sign up for a hosting account, your hosting provider will give you two name server addresses (the primary and secondary) that you will tell the registration company where you want your domain to point to, i.e., your hosting company.

Domain Propagation
There are various name servers set up throughout the internet that all sync up when changes are made, a process known as propagating. Once that propagates through (it can take up to a day in most instances) then when someone types your new domain into their web browser, it will look up where to go based on the domain name server.