There are organizations that are called registries that manage each TLD option. When you register domain names, you would first visit a domain registrar who would in turn check with the registry that manages that TLD to see if the domain is available. If the domain for the TLD you want discovers that it is in fact available, the registry adds it to their registry and gives you access to it. If the domain has already been registered by another user for that TLD, you will be shown during the domain registration process that the domain isn't available.
Domain names are often seen in analogy to real estate in that domain names are foundations on which a website can be built, and the highest quality domain names, like sought-after real estate, tend to carry significant value, usually due to their online brand-building potential, use in advertising, search engine optimization, and many other criteria.
It is also possible to setup a personalised or branded email address that simply forwards email to an existing inbox you may already have elsewhere such as Gmail, Yahoo or similar. This way, you can continue using your existing email address and receive your emails through your branded domain name. To send from your personalised address you will need an email mailbox account.
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.
Namecheap is a solid choice for registering your domain name. Plus, their site is incredibly intuitive and easy to use, especially on their domain management pages, which can be incredibly helpful. They offer reasonable priced domains, and have a free DNS service, and WHOIS protection. They also offer SSL encryption, for those looking to beef up the security of their domains.
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.
Every time you visit a website, your computer performs what's called a "DNS lookup". When you manage a domain with us, you get to use the same DNS servers as Google. This means your domain name will connect quickly and reliably to your website. We include 10 million resolutions per year for each domain you register with Google Domains. Learn more in the Help Center.