Some companies will provide both web hosting services and domain name registration, but there's no reason you have to use the same company for both, though it's often more convenient to do so. When you register a domain name with almost any service, you have the ability to easily set what is known as the nameserver information, which basically tells traffic where to go to get to your website, so you can change your webhost quite easily. You can also transfer your domain registration between companies (usually at no charge), so whatever you do doesn't lock you in forever.
You don't need to have a website to register a domain name, so if you have a great idea for name, but you don't have a website or even a plan for how to create a website, it's still not a bad idea to register the name now. It will cost less than $10/year to get the registration, and that means nobody else can register your name while you're working on your site. If you later decide not to bother with a website, you can let the registration lapse and you've only lost your registration fee.
There are hundreds of companies which will allow you to register a domain using your credit card. It takes only a minute to do. Various companies offer various levels of service and options. Some allow you to forward traffic and email sent to your domain to an existing website or email account. If you don't want everyone to be able to look up your name by looking at the registration information for your website name, some companies will "cloak" your identity. Some charge a free, other's don't.
So check around and see what's on offer today. For this website I transferred my domain name to my hosting company HOSTMONSTER . As I said I could have my domain registered with one company and my website hosted by another, but it's a bit more convenient to have them both in the same place. In addition HOSTMONSTER include free registration in their hosting package, so I only have one bill to pay (for hosting) and my domain gets renewed each year at no extra charge. HOSTMONSTER offer free cloaking of your name, address and phone number if you want to keep them private.
You can also register domains and get hosting packages through sites like DOTSTER.COM and GODADDY.COM , both well established companies with a good reputation. Expect to pay something between about $6.95 and $15.95 per year for .com domain registration. The price depends on exactly what services the domain registrar offers and what extension (.com, .net, .org, .info, .biz etc.) you want.