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".
GoDaddy used to be a place I'd recommend only as a joke, but has significantly improved their service. They'll recommend you domains based on certain keywords, which is useful if you've not got the one you want available (they'll actually remove these from the search to try and not break your heart), and they've also significantly improved their hosting offering. They still continue to offer weird things like "email services" that other hosting companies provide too. Our GoDaddy Review.
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.
Premium domains (also known as aftermarket or pre-registered domains) are short domains, often just one word or even just 3-5 letters. Most of them have a .com extension but many premium domains end with .org, .net, and .biz. These domains include common words and are generally the most memorable. Because companies value short domains that match their company name or products, these domains are typically the most desirable. Additionally, certain domains sold by different registries are considered premium and therefore have a higher price point. In some cases, the renewal costs of these higher-priced domains are also quite expensive. Some high-priced premium domains, though, renew at a regular (lower cost) rate, giving you a better value in the long term. Make sure to research the overall cost of the domain plus renewal to find the right domain for your budget. You can browse Namecheap's premium domains in our Marketplace.