If you ever change your web host (the actual physical location of your website files), your domain name goes with you. Your regular visitors or customers who knew your site name as (say) www.thesitewizard.com would not have to be informed about a change of web address (also known as "URL"), since as far as they are concerned, the site is still at the same place. They would simply type your domain name and they'd be transparently brought to your new location.
Domain name registrars, in case you didn't know, are the companies that manage all the website domain names you'll see (EG: Inc.com) - the fingerprint of your business, which may also include the hosting and privacy measures. For example, by default many companies will actually over-price the simple process of keeping your registration information private (which if it isn't can be found via a simple Whois search).
Note that you can also transfer your domain name from one registration service to another. You'll want to do this if you're not satisfied with your current domain hosting service, if you find a better deal when your current registration is coming due, or, most likely, if you've signed up with a web hosting service that will also transfer your name to its site. Expect to get the transfer for free, but if that isn't offered, search for another domain hosting service.

Premium domains (also known as aftermarket or pre-registered domains) are short domains, often just one word or even just 3-5 letters. Most of them have a .com extension but many premium domains end with .org, .net, and .biz. These domains include common words and are generally the most memorable. Because companies value short domains that match their company name or products, these domains are typically the most desirable. Additionally, certain domains sold by different registries are considered premium and therefore have a higher price point. In some cases, the renewal costs of these higher-priced domains are also quite expensive. Some high-priced premium domains, though, renew at a regular (lower cost) rate, giving you a better value in the long term. Make sure to research the overall cost of the domain plus renewal to find the right domain for your budget. You can browse Namecheap's premium domains in our Marketplace.
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