The domain name is the beginning of everything. Exactly like a car tag, it identifies your website, it is the address to your site. There are 3 things in use when it comes to having a website; The domain name, the hosting services, and the website. if you loose the information to your hosting service, and your website becomes accessible, then all yu need to do is simply disconnect your domain name from that hosting company name servers and move on to another hosting company. Of course you will have lost your website and unless you have a back up, you may end up having to design a new site.
But as long as you have your domain name you will be in control. Now, why is the domain so important that you should be the one to register it? Let say that you build a website, you upload it to a server under a domain name. A few months passed and suddenly your domain starts to become popular. Domains can be sold and some domains based on how popular they are can be sold for thousands of dollars. Not only that, if you are receiving profit through your domain and it was register by someone else, this person holds all the rights to the domain which means, he can get away with everything and you end yup getting nothing.
Just like the title of a car.