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.
Registries and registrars usually charge an annual fee for the service of delegating a domain name to a user and providing a default set of name servers. Often, this transaction is termed a sale or lease of the domain name, and the registrant may sometimes be called an "owner", but no such legal relationship is actually associated with the transaction, only the exclusive right to use the domain name. More correctly, authorized users are known as "registrants" or as "domain holders".
I’ve tried to be charitable to Verisign in my reading of Site-Finder-gate. Even then it seems quite clear that they were deep within the grey areas of their contract with ICANN, and that they knowingly acted against the spirit of the Internet that they’d been entrusted to build. That said, I respect the hustle — Verisign were, and continue to be, a publicly traded company with a responsibility to deliver value to their shareholders, and this quasi-bait-and-switch was a pretty inspired way to do it.
The way that competition is supposed to work between registries, is that lots of companies go to ICANN and say “We think we can reliably manage .com, and we think it would cost us [$5] per domain.” ICANN reviews these bids and awards the .com contract, which lasts for a fixed period of time, to whomever they deem best-suited to the job. In this time, other companies can think of ways to manage .com better/cheaper, and try and win the next contract. This, in theory, would provide an incentive for registries to innovate, a big part of which would be in lowering costs.
A domain name consists of one or more labels, each of which is formed from the set of ASCII letters, digits, and hyphens (a-z, A-Z, 0-9, -), but not starting or ending with a hyphen. The labels are case-insensitive; for example, 'label' is equivalent to 'Label' or 'LABEL'. In the textual representation of a domain name, the labels are separated by a full stop (period).
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.
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.