In effect you will be renting this name from "The Internet", since you'll have to pay a fee to keep the name current. The more years you rent, the better deal you can get. A number of companies are authorized to rent you a name, but the Domain Name is good over all the internet. The companies that perform this rental service are called Registrars. When you register your name with a company, for example GoDaddy, or Network Solutions, they will allow you to manage your name through your browser. You provide billing info, and contact info, and pay the fee. You can log in through the Registrar's management interface, with your domain name and password, inspect your payment situation, and change management information, pretty much whenever you want.
What is Domain Name management? It's keeping your rent current, your contact and billing info current, and crucially telling The Internet where to find your web pages (and other services). The Internet rests on Domain Name Services, DNS. YOU get to tell the net where those services are located. The location takes place in "nameservers". There will be a primary nameserver, and a backup nameserver, and they'll have the same information. What you provide are two nameserver addresses that your hosting company will tell you to use. The addresses will look something like this: "ns6259.hostgator.com", or perhaps like "206.127.64.240". Log into your registrar and put in what your hosting service tells you. Voila, after this information percolates through the net, the connection is made, you're on the net.
Hosting companies will provide many services, among them storing your web site pages and providing them to browsers whenever anyone types http://yourDomainName in their browser. Another service a hosting company will provide is email connections and storage.
Registrars typically also provide hosting. Being a little paranoid, i like to manage my domain name with one company, and rent my hosting from another. As long as you pay the domain name rental, there isn't much need to move that management around. Oh yes, once you have control over a domain name, it is "yours" to the extent that you can move the management to another Registrar. Seldom is there a need to do that.
Web hosting, on the other hand, is a very rapidly changing business. You may very well want to start with one hosting company, and switch to another at a future date. Additional complications are, for example, selling your business and the domain name that goes with it. You have easier control if the domain name management is the only thing your registrar is providing.
Changing web hosting, an Example: You have your domain name management with registrar "Reg". You are hosting your web site at company "Ahost", but you find a better deal with the expanded services you crave with hosting company "Bhost". While Ahost is serving your web needs, you build your new site at Bhost. That takes a little while, but when it is done, you log in to Reg and change the two nameserver entries to the ones that Bhost provides. As soon as the nameserver entry changes cascade through the internet, all the traffic to your domain name goes to Bhost. (There would be a period of less than 24 hours where both Ahost and Bhost were serving, tricky if you're a big company, maybe trivial otherwise.) Turn off Ahost and the switch is done. If you have Ahost as your registrar, things might be more complicated.
