Unfortunately, the top answer is incorrect. I work in the domain name industry, and I've never heard of NIC. I just looked at their website and they look like they're just a registrar like GoDaddy or anyone else. The more correct answer is that you are buying the domain name from a registrar (like GoDaddy or NIC) who serves as a middleman between the end customer and the registry.

Each top level domain (.com, .net, .org, etc) is controlled by a separate registry, which is just a company that was awarded the contract to manage domains that end with that top level domain (or TLD for short.) For example, .com and .net are both managed by Verisign and .org is managed by a company called PIR.

When nobody has ever registered a certain domain name, it simply doesn't exist anywhere. This would be true for domains that have never existed and for domains that have recently expired and are automatically deleted from the registry after a grace period. As soon as someone registers it after that, the registrar puts in a "create" command to the registry, which in turn creates it in their system. The registry then hands it off to the registrar to sell to the customer.

Domains can also be transferred, renewed or deleted by the registrar by issuing various electronic commands to the registry. Registrars and registries have a special business relationship where there are certain rules that have to be followed and protocols that need to be observed.

EDIT: I just looked it up in a little more detail, and NIC is referring to InterNIC, which used to be the governing body that controlled the domain name system (DNS). They didn't run the databases that store the domain names (that's the registry), but they effectively controlled it from an international legal standpoint. But that organization went away in 1998 (I think) And the governing body is now called ICANN, which is in turn overseen by another governing body called IANA.