If I go to "http://webmasters.stackexchange.com," how does my browser know how to find the server for "webmasters.stackexchange.com?" It knows through using DNS lookups, which means it needs to ask a DNS server if it knows the address of webmasters.stackexchange.com.
So who owns those DNS servers? Typically, it will be your Internet provider; they aren't personally responsible for knowing the address of every web site across the 'Net.
So who is responsible for telling all DNS servers where web sites live? Domain name registries are. Somebody needs to be an ultimate authority on addresses for domain names; if nobody was, then anybody could pretend their servers go to your domain name. Obviously, managing domain names takes support people, developers to give webmasters the tools for managing domains, servers, and many other things that cost money.
That's why you have to purchase a domain name, so you can claim ownership and pay for the costs of the services provided by the registry. However, there are services, like DynDNS, that let you buy (or use for free in a limited fashion) a sub-domain under a domain name that they own.