There are numerous domain name registrars. Listed below are just a few, along with my comments, if I know anything about them. Note that the domain name industry is highly competitive, with prices wildly fluctuating throughout the year, every year, so it's impossible to really mention accurate prices below unless I spend all my time updating this page. Please check their sites for the latest rates. All prices below are in US dollars.
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In the first quarter of 2015, 294 million domain names had been registered. A large fraction of them are in the com TLD, which as of December 21, 2014, had 115.6 million domain names, including 11.9 million online business and e-commerce sites, 4.3 million entertainment sites, 3.1 million finance related sites, and 1.8 million sports sites. As of July 2012 the com TLD had more registrations than all of the ccTLDs combined.
The hierarchy of domains descends from the right to the left label in the name; each label to the left specifies a subdivision, or subdomain of the domain to the right. For example: the label example specifies a node example.com as a subdomain of the com domain, and www is a label to create www.example.com, a subdomain of example.com. Each label may contain from 1 to 63 octets. The empty label is reserved for the root node and when fully qualified is expressed as the empty label terminated by a dot. The full domain name may not exceed a total length of 253 ASCII characters in its textual representation. Thus, when using a single character per label, the limit is 127 levels: 127 characters plus 126 dots have a total length of 253. In practice, some domain registries may have shorter limits.
If you already have a web host, obtain from them the names of their primary and secondary nameservers. Don't worry if you don't understand what these things mean. Just save the information somewhere. The information can usually be obtained from their FAQs or other documentation on their site, usually under a category like "domain name" or "DNS" or "domain name transfer" and the like. If you can't find it, email them. You'll need the information to point your domain name to your website after you buy your domain. Having said that, if you don't have a web host yet, don't worry. Just read on.
Sedo is where I bought my domain - over Escrow - and is a global marketplace to buy, sell, and park domain names. They've over 18 million domain names for sale, but the big part is that they're a huge, well-respected company. This may not be important if you're spending $150 on a domain, but in a $15,000+ purchase - which happens in many cases - you will be potentially buying from a foreign entity, that you don't know, and many of them will suggest an escrow payment. This means you put the money in an account owned by a third party that will hold it and only pass it on once the domain has been transferred over into their name (which in this case is what Sedo does). This can be quite disconcerting - but Sedo is well-known and has done many, many deals - and the process is about as full of hand-holding as handing tens or hundreds of thousands of dollars for a domain name can be.
A fully qualified domain name (FQDN) is a domain name that is completely specified with all labels in the hierarchy of the DNS, having no parts omitted. Labels in the Domain Name System are case-insensitive, and may therefore be written in any desired capitalization method, but most commonly domain names are written in lowercase in technical contexts.
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.