Domain name registrars, in case you didn't know, are the companies that manage all the website domain names you'll see (EG: Inc.com) - the fingerprint of your business, which may also include the hosting and privacy measures. For example, by default many companies will actually over-price the simple process of keeping your registration information private (which if it isn't can be found via a simple Whois search).
There are organizations that are called registries that manage each TLD option. When you register domain names, you would first visit a domain registrar who would in turn check with the registry that manages that TLD to see if the domain is available. If the domain for the TLD you want discovers that it is in fact available, the registry adds it to their registry and gives you access to it. If the domain has already been registered by another user for that TLD, you will be shown during the domain registration process that the domain isn't available.
FatCow has one of the cheapest offers for an entire website package, with hosting and domain names for just thirty cents during your first month. After, that the price is $10.99 a month or $59.88 for the whole year, in a shared hosting package with enough bandwidth for most people. This is a limited promotion I found, but they still sell fairly well-priced packages (their "fat" options come with extras like marketing and advertising credits), and will gladly offer you perks for transferring your domains and hosting to them.
The first step to starting your website is registering your domain name. Think of your domain name as a street address for your website. Without a domain name, you would have to tell customers to visit your website at a temporary url such as 220.127.116.11/~yourwebsite instead of yourwebsite.com. Get your own domain name to lend your site a professional look and establish your online brand!