Domain Name Registration HOWTO

A computer with a static IP address and permanent connection to the Internet

This can be difficult to run by yourself; make sure someone with the proper skills is in charge of administration. Once you have the computer, the address and net connection are the only things you absolutely must pay for. Most people pay a "web hosting provider" to own and operate a machine and connection on their behalf.
Cost varies according to bandwidth requirements.

Two seperate machines, also with static IP addresses and permanent net connections

These will act as "name servers", telling the rest of the Internet what number to use when someone or something comes looking for your domain name. Also somewhat difficult to run by yourself.
Many hosting providers offer their own name servers to customers as part of a package.
Minimum non-free cost: Approximately $3 per month, averages $5 per month.

The domain registration

Put up any web pages you wish to be available when your domain becomes active. Create all email addresses you wish to be able to receive mail at your domain.
Tell your name servers the static IP address of the machine which hosts your domain, unless your provider is handling this.
Register the domain with a registrar.

In addition to the registration fee and required contact information, you must also provide the registrar with the names and IP addresses of your two name servers.
Minimum non-free cost: Approximately $10 per year (less than a dollar per month!) for "traditional" domains such as .com, .net and .org. "Country code" domains (.us. .uk) range from free to very expensive.

It usually takes a few days for a new address to spread to the rest of the net and everyone can "see" it.

All of these can in theory be gotten free except for the physical hardware and net connection if you're willing to live with the relevant disadvantages. However, it seems that with the newly competitive market, complete control of one's own domain is well within the average budget of most individuals.