In your capacity of the website owner, it would be your attempt to register a domain name which would also be the name of your website. At a later stage this would make it easier for the visitors to find your website. But, given that there are millions of websites out there on the internet; it may be difficult to find success in such a search.
You may consider generic names like cars, automobiles, planes, toys. But here again the chances are that they may already have been taken. To continue the search for the domain name in this fashion, you may also consider hyphenated names; but here the danger would be that the visitor may not apply the hyphen or not remember to apply it in his subsequent visit. Another way to resolve this issue of domain names is to check whether the domain name with a .net or .biz extension is available. Of course, the top level domain name is the .com. If none of the extensions are available for the domain name you have selected for your website; you may consider contacting the owner of the website/domain name and offer to buy it at a fair price. The worst that can happen when asking to buy is that the present owner would refuse to sell.
Moving along this path of getting the domain name for your website, you may consider the plural form of the name. Here it would be useful to remember that visitors tend to remember short domain names; which of course, are not likely to be available. On the other hand a longer domain name may be useful as far as keywords are concerned; which in themselves would be of some importance when it comes to the matter of search engine algorithms.
Other aspects related to your search for an appropriate domain name would include, whether your webhost offers multiple domain hosting; which would enable you to have more than one domain name or website in your server account. While some webhosts plans would offer three domain names per account other may even offer five. The advantage of this would be that you would be able to resell the remaining (or unused) two of three or four of five domain names.
There may also be a situation that, you have decided upon the name of your website which is yet to be developed. In which case you would purchase the domain name from a domain registrar, and have your domain name placed on a name server to be used later. This is known as domain parking, and has the main advantage of preventing others from acquiring that domain name.
There is also the question of sub-domains to be considered. These are third level domains, which you would utilize with the express purpose of organizing the content of your website. Sub-domains are folders in the root directory of your server. And to access them, the visitors would have to prefix the sub-domain name (which would replace the www in the URL) to the domain name while typing the name in the address bar of his browser.
The domain name would be registered and belong to you the moment you have paid the registration fee for it for the first year. Thereafter, you would be required to pay an annual renewal fee for the continued registration of the domain name in your name. There may be occasion when you would like to pay this renewal fee for multiple years; in which case the domain name registrar is expected to give you a discount on the registration/renewal fee.
In case, for reasons beyond your control your domain name has expired; you may renew it within a period of one month from the expiry date. If this is not the case, then the domain goes into what is called the redemption phase. Your domain would remain in this redemption phase for a period of 15 to 30 days, before it is made available for purchase by other website owners and webmasters. To get a domain out of the redemption phase can be an expensive exercise; and to cut a long story short, it would be best to pay the renewal fee for your website before or on the expiry date. Usually a renewal due email is sent out by the service provider well in advance of the expiry date of your website's registration. Further, to be able to check the renewal due date or to confirm that your website domain has been renewed you could do a Whois search for it, or visit domaintools.com and check the particulars of your website listed there.