The terms are being thrown around here a little loosely so let me repeat what I said above and hopefully it will make more sense this time.
HOSTING companies do not register domain names. REGISTRARS do. If you want to buy (register) a domain name such as www.yourdomain.com then what you do is first go to a REGISTRAR that is authorized to do so by the internet authority; such as GoDaddy or similar.
After registering your domain with a registrar then you go to a hosting company and buy some hosting ... if you want to. You can theoretically host it yourself on your own PC but I have done that and it is a pain and far beyond the capabilities or hassle level of the average user. The hosting is where that actual website files will reside and be served up to any visitors that visit your domain.
The reason that this can be confusing is that often times registrars also sell hosting and hosting companies can be authorized registrars too. But these are two separate things. You can have one company as your registrar and another as your hosting.
Think of it like a car. You have license plates and the car itself. The dealer can get your plates for you but that does not make him DMV. And you can buy a car from the state (like at an auction) but that does not preclude you from going to DMV and getting plates. And in order to drive your car on the street you need BOTH a car AND plates.