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1) Make up a name, and check to see if it's already in use.
2) Find a registrar. Here's a long list of Internic accredited ones. Make sure the registrar provides it's own DNS service and web front end.
4) Pay the registrar money.
5) Find the IP address of the hosting computer, and create an 'A' record that points to it.
6) Wait for the change to propigate through DNS (anywhere from half an hour to half a day, depending on the phase of the moon and color of your underwear).
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you go to http://billing.webspacesolutions.com/order
put in your desired domain name, click next, if its available click continue, select a package click continue, fill out all the rest of the information, once the order is complete you will be emailed an automated welcome email with your username/password, and ftp information :)
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There are a two separate services: registering/controlling the domain name and having web hosting of some sort behind it. For example, buying "phallicimage.com" just gives me the ability to make the name refer to a computer of my choice. I think 1 & 1 is one of the cheapest ($6/yr) and they don't hassle you the way other some do.
Obviously, being able to use the domain name to refer to a machine isn't terribly useful if you don't have a machine to host your site/e-mail/ftp/whatever. This part is called "hosting."
There's a plethora of companies providing either service, and most offer both. I've used Host-Cat, FuitadNET, and 1 & 1. If you want to find more search for "web hosting" or "domain registration."
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http://www.register.com
http://host-cat.com/
www.freewebs.com
http://fuitadnet.com/prodserv.php#domain
http://order.1and1.com/
https://www.godaddy.com/gdshop/default.asp