A handful of domains will have restrictions on them, which means you can only purchase them if you meet certain criteria or have authorization (some examples are .gov, .edu. and .mil). But most extensions are available to everyone. In fact, most country code Top Level Domains (ccTLDs) are available for anyone to purchase, even if you don’t reside in the country in question.
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.

With over 25 million .com domains registered with Google alone, this top-level domain is by far the most popular choice worldwide. According to the domain marketplace Sedo, the average sales price for a .com domain name during the second quarter of 2015 was 4,701 dollars. The most expensive publicly traded domain names have been known to fetch eight-figures. But don’t quit your day job just yet: such sales are the exception rather than the rule.
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.

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.

A domain name can serve as your distinct identity on the internet. If you want to promote yourself, your brand, or your business, a specialized domain name is essential. A domain that is tailored to your needs will make you look professional and capable and can boost your presence on the web so you get noticed. If you’re starting a business, a domain name search can help you become discoverable and accessible. We’ve got some fantastic tips to help you discover the most suitable name for your needs.
Now it’s time to see if your domain name choices are available. Once you’ve come up with a few strong domain candidates, it’s time to plug them into Namecheap’s domain name search bar. We will search available domain names for you and let you know if your first choice of second-level domain or TLD is available. We’ll also suggest some alternate TLDs you might try if your first domain choice isn’t available.