1&1 Internet — This is primarily a large web host that is also a domain name registrar. Like all registrars, there are different charges for different domain suffixes. For example, at the time I last checked, you pay $0.99 for a ".com" on your first year, then $14.99 per year thereafter. The fee includes private domain registration, which means that your particulars are hidden from public view (done by registering the domain in the name of a proxy company). You also get a free email account and unlimited email forwarding, DNS management, a free SSL certificate for your domain, etc. Both credit card and PayPal payments are accepted by this registrar.
A few companies have offered low-cost, below-cost or even free domain registration with a variety of models adopted to recoup the costs to the provider. These usually require that domains be hosted on their website within a framework or portal that includes advertising wrapped around the domain holder's content, revenue from which allows the provider to recoup the costs. Domain registrations were free of charge when the DNS was new. A domain holder may provide infinite number of subdomains in their domain. For example, the owner of example.org could provide subdomains such as foo.example.org and foo.bar.example.org to interested parties.
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.

GoDaddy — This extremely popular registrar (probably the biggest today) offers .com domain names for $9.99 (plus 20 cents) per year (or $6.99 plus 20 cents if you transfer from another registrar). They have a web interface to manage your domains, free web redirection (where people who visit your domain will get transferred to another web address of your choice), free starter web page, free parked page or free "for sale" page, and an optional private domain registration where your domain is registered in the name of a proxy company. Besides .com, they also sell .us, .biz, .info, .net, .org, .ws, .name, .tv, .co.uk, .me.uk and .org.uk, etc. Note that (as with all registrars) the exact price varies depending on which domain you are registering (for example some domain suffixes are more expensive than others). Both credit card and PayPal payments are accepted.
iPage's web hosting packages start as low as $1.99 per month and they throw in domain registration for free during the first year, which makes them a bargain for very, very lean startups or one-person operations. They also have the classic domain name search portal, and provide a tutorial on the new TLDs that helps, considering a new one seems to be appearing every second or two.

Even when you register and pay for your domain name, you won't necessarily be able to use the name for several hours or even a few days. The domain must propagate, meaning that the official domain name registry must be updated with your website's Domain Name System information. That's something that occurs on the backend without any need of input from you.
Domain names serve to identify Internet resources, such as computers, networks, and services, with a text-based label that is easier to memorize than the numerical addresses used in the Internet protocols. A domain name may represent entire collections of such resources or individual instances. Individual Internet host computers use domain names as host identifiers, also called host names. The term host name is also used for the leaf labels in the domain name system, usually without further subordinate domain name space. Host names appear as a component in Uniform Resource Locators (URLs) for Internet resources such as web sites (e.g., en.wikipedia.org).
A great domain name is a concise, easy to type, and memorable URL that reflects your brand or your website’s subject matter. Avoid using hyphens, strings of numbers, or unnecessary words to make it easy for your visitors to remember and find your website. Remember: A great domain name is one that your visitors can type correctly on their first try.
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