If possible, you want your domain name to be the same as your website name, which should probably be the same as your company name, with a ‘.com’ extension if you are a business. However, the chance of that exact name and extension being available may not be good.
Start by searching for available domain names that would work for you. Domain Tools is a great place to begin your domain name search. If you can’t find the exact name you desire, consider using hyphens for a multiple word name, or an abbreviation of your company name, like an acronym, or using a different extension, such as ‘.biz,’ ‘.net, .’org,’ ‘.info’ or ‘.us.’
There are several more domain extensions to choose from, and one of them may be more appropriate for your particular needs. Here is an article at thesitewizard.com that goes into more depth about Choosing a Good Domain Name.
While you may want to register your domain name with the same company that will host your website, getting the ‘right’ domain name is often so much more difficult that it’s best to register one as soon as possible. You shouldn’t pay more than about $30.00/year for a domain name, and many companies will offer domain registration for as low as $7.00 or $8.00 per year.
Here are links to three hosting services with whom I have had good luck where you can register your domain name for $15.00 per year or less:
If you want to understand domain names at a more basic level, this article at theinternetdigest.net explains what they are, how they came to be, and more.