Think of your domain name as a street address for your website. Without a domain name, you would have to tell customers to visit your website at an IP-based url such as 18.104.22.168/~yourwebsite instead of yourwebsite.com. The former is not very memorable or friendly to human eyes and will surely result in not that many visits to your website. While it is not mandatory that you have a website domain name, nor is it mandatory that a domain you own be attached to a website, for a successful online presence you truly will want to get your own domain name to ensure your website has the proper air of professionalism and appropriate branding.
A handful of domains will have restrictions on them, which means you can only purchase them if you meet or exceed 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 code in question.
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