Think of a few good domain names that you'd like to use. It won't do to think of only one — it might already be taken (it probably is!). You can find some tips on choosing a good domain name from my article, Tips on Choosing a Domain Name, at http://www.thesitewizard.com/archive/domainname.shtml

There's more to it than meets the eye.

You will need either a credit card or a PayPal account to pay for the domain. This is a requirement of most if not all registrars. It will allow you to claim and get the domain name immediately on application. This is not an option (unfortunately).

If you already have a web host, obtain from them the names of their primary and secondary nameservers. Don't worry if you don't understand what these things mean. Just save the information somewhere. The information can usually be obtained from their FAQs or other documentation on their site, usually under a category like "domain name" or "DNS" or "domain name transfer" and the like. If you can't find it, email them. You'll need the information to point your domain name to your website after you buy your domain. Don't worry if you don't have a web host yet. Just read on.

If you do not have a web host, you can always allow the registrar to park your domain name at a temporary website specially set up for you. This way you can quickly secure your domain name before it's too late and still take your time to set up the other aspects of your site. As far as I can tell, most registrars (or probably all) automatically park your domain by default whether you ask them to or not, so if this is your situation, you probably don't have to do anything special to get it done. Some of those registrars also provide you with a free email address at your own domain name while it is parked at their site, like sales@example.com (where "example.com" is your domain).