Here is the history of ICANN which took over control of domain name registration in 1998: http://en.wikipedia.org/wiki/InterNIC
A domain name is leased.
If the domain name that you want is available then you can lease it.
For example if you wanted the domain "myonlinestore.com" then you could register the exclusive use of that name for around $15 per year.
When you register a domain, you get control of the location that visitors are sent when they type in your domain name. You could redirect them to an existing website, or send them to a new site. This is done by setting the nameserver configuration where you registered the domain.
No domains are completely free.
Many web hosting packages include a free domain name registration as part of the cost of the package. Be sure to read all of the details on the various hosting packages as some have costs that rise after the promotional period expires.
As far as expensive domains go,
Domain names can be registered for up to 10 years in advance with the option to renew the name indefinitely. For Example, If you have registered a available domain name for $15, you have the option to resell the rights to that particular name for whatever price that the market will allow. Insure.com sold for 16 million in 2009. Here is a list of the most expensive domain names sold. http://most-expensive.com/domain-name
For a business a easy to type and remember domain name is vital. However smart entrepreneurs think outside the box. Who knew what a "google" or "yahoo" was before they became popular. Take a look at the list of most popular websites. http://www.ebizmba.com/articles/most-popular-websites What is a "bing" or "twitter", I think you get the point.
Good Luck...