If you want a domain name to remain your property, you need to pay a yearly fee, which is usually quite reasonable, but can vary from one extension (e.g. .com, .net, .org, .info, .co.uk, .us, .tv...) to another, and also from registrar to registrar. The payment can be done for several years in advance (usually up to 10 years). Each extension is under the supervision of a registry; in most cases, you don't pay the registration and renewals directly to the registry, but to a registrar of your choice. Some registrars offer only a few extensions, other ones dozens of them.
In case you do not already have an account with a registrar, you should first open one: very easy to do, and no need to have a domain first. Among US-based registrars, Name Tactic recommends especially Dynadot, NameSilo and Uniregistry.
In case you buy an already existing domain name from Name Tactic or from another domaining service, it might be a good idea to open an account with the same registrar as the seller, which will allow in many cases a fast push rather than a transfer.