If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.
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.

The practice of using a simple memorable abstraction of a host's numerical address on a computer network dates back to the ARPANET era, before the advent of today's commercial Internet. In the early network, each computer on the network retrieved the hosts file (host.txt) from a computer at SRI (now SRI International),[4][5] which mapped computer host names to numerical addresses. The rapid growth of the network made it impossible to maintain a centrally organized hostname registry and in 1983 the Domain Name System was introduced on the ARPANET and published by the Internet Engineering Task Force as RFC 882 and RFC 883.

For the most part, you can pick whichever TLD you prefer, though a few have restrictions -- you can't buy a .edu domain unless you run an accredited post-secondary school. Many, but not all, country-specific domains (.us, .uk, .jp) are restricted to citizens of the country in question. Unrestricted country domains are often used for "domain hacks," where the extension becomes part of the title, such as YouTube's alternate address, "youtu.be."
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DNS is essentially a database of IP addresses.  Each IP address is a series of numbers that allow computers to communicate with each other. A website is identified on a server by its IP address. Instead of your visitors having to remember an IP address to type into their browser to access your site, they instead use your domain name. When your domain is entered into their browser, DNS translates it into an IP address because that is what computers understand.Your computer is able to connect to your web hosting and show your website on the browser once the IP address is found.

ICANN publishes the complete list of TLD registries and domain name registrars. Registrant information associated with domain names is maintained in an online database accessible with the WHOIS protocol. For most of the 250 country code top-level domains (ccTLDs), the domain registries maintain the WHOIS (Registrant, name servers, expiration dates, etc.) information.
Not every registrar sells every top-level domain, so if you want an uncommon TLD like .pizza, you might have to try a few registrars. Unusual TLDs often cost more than a standard .com address, but provide built-in memorability and give you more name options, since far fewer .pizza addresses are already taken. You can check name availability at any registrar that offers the TLD you want.
Domain names were created to make IP addresses easy to remember. Every computer has an IP address assigned to it – much like a street address. But instead of having to memorize each IP address number, we assign domain names to these numbers so we can easily remember them. The domain name system, or DNS, takes domain names and translates them into their IP addresses so that computers can communicate with one another. When you enter a domain name into your web browser, the browser simply uses that domain to locate its corresponding IP address and then shows you the website associated with it. If you own a business, GoDaddy can help you find a domain name that best suits the business so you can easily promote your website.
Don’t get discouraged in your domain name search if what you’re looking for is already taken. We have a couple of ways for dealing with such a situation. You can try our Domain Buy Service, where we negotiate acquiring the domain from its current owner. We also offer a Domain Backorder service, where you purchase a backorder credit to give you a chance to get the domain when it goes into auction.

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.[2]