How do you do this? You go to a company called a domain name registrar, and you pay them to make sure a particular name is registered with you. You become the owner of that domain for as long as you’re paying for it, however, the terms of the contract work out.
A domain name registrar is exactly what it sounds like: it’s an organization that manages the registration of domain names. To be official, a registrar has to be accredited by a generic top-level domain registry—don’t confuse the two—and a registrar will operate in accordance with the registry.
Registries are an article for a different time, but briefly: registries are databases of domain names and all the information of the people who registered them. Registries are managed by an organization called ICANN (Internet Corporation for Assigned Names and Numbers), which is a huge deal for domain registration.
Basically, you can only register from companies that have been accredited by ICANN. This won’t narrow things down for you, because there are still a lot of ICANN-accredited options.
So now we know what a domain name registrar is.