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.
If you have a web site, you should seriously consider registering your own domain name. A domain name is a name like "thesitewizard.com" or "thefreecountry.com", which you can use to refer to your website. Note that you do not have to be a company or organisation ("organization" if you use a different variant of English) to register a domain name. Any individual can do it too.
My personal preference is to register the name directly with a domain name registrar rather than through my web host. I've heard stories, in the past, of less-than-reputable web hosts that registered the domain under their own name, making them the owner of the domain rather than you (although I don't know if such web hosts still exist today). Registering direct with a domain name registrar allows me to make sure that I am registered as the owner, the administrative and technical contacts. Being the owner is vital — if someone else places himself as the owner (such as your web host), he can always decide to charge you some exorbitant fee for the use of the name later, and there is little you can do. The various other contacts are less vital, but may still play important roles, depending on your registrar. For example, for some registrars, the administrative contact's approval is required before a domain name is transferred out of a web host (or at least, it used to be). If he/she cannot be contacted, the technical contact is used.
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The way that competition is supposed to work between registries, is that lots of companies go to ICANN and say “We think we can reliably manage .com, and we think it would cost us [$5] per domain.” ICANN reviews these bids and awards the .com contract, which lasts for a fixed period of time, to whomever they deem best-suited to the job. In this time, other companies can think of ways to manage .com better/cheaper, and try and win the next contract. This, in theory, would provide an incentive for registries to innovate, a big part of which would be in lowering costs.
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.