A domain name is usually $4-$9 at various internet registrars. You usually fill out a form with your name, and their service tells you (instantly) whether that name is available or not. When you find a name that's available, you can go ahead and pay for it online (with a credit card...though some places also take echecks or paypal). I recommend GoDaddy...they're the registrar I've been using for years.
Whichever registrar you use, it's vitally important that your domain is registered in your name. This means that the legal owner of the name must be either you or your organization. No exceptions. No matter what. Ever.
Failing to ensure this will lead to a situation where your domain will be legally someone else's property...perhaps belonging to an unscrupulous registrar, or to a shady website designer. GoDaddy registers all domains in your name, and you have the option to designate your organization the legal owner, if you wish.
There is never a good reason for your domain name to be owned by someone else...no matter what. If your website designer tells you he needs everything registered in his name because it'll help stop global warming, find a new website designer.
(Incidentally, if you're looking for a website designer, I'm always open to new projects...send me an email and we'll talk).
Once your domain is paid for, you'll usually be sent an email with information on how to log in and change the settings associated with your domain. I usually avoid all the special offers and other junk they try to sell you (domain locking, email accounts, hosting, and a million other things), but if you signed up for something, you'll probably have some options to play with for those as well.
The only thing you really need to do at this point is set your nameservers and fill in any missing contact information for the domain, and you're done.
