Once you've chosen which of the Domain Registrars you're going to use, you simply sign up for a free account on their site, fill out their online domain registration form, and pay for your domain.
Woo Hoo! You've just completed a major step on your path to authority site success!
Unfortunately, there are a few unscrupulous people out there who attempt pirate other people's domains or interfere with their online businesses. So you may want to pay your domain registrar an extra fee for domain privacy (so your name and contact info are kept confidential from people who look up your domain's info), and have your domain registrar lock your domain name.