Getting a domain name involves registering the name you want with an organisation called ICANN through a domain name registrar. For example, if you choose a name like "example.com", you will have to go to a registrar, pay a registration fee that costs around US$10 to US$35 for that name. That will give you the right to the name for a year, and you will have to renew it annually for (usually) the same amount per annum.
There are tons of great domain names available for registration and it can take as little as five minutes to register one at any domain name retailer. These online stores make it very easy for you to search for available domain names with name suggestion tools. They also offer complementary services, such as Web hosting and website design so you can register your desired domain name and set up a website all at once.
Domain name registrations are priced on a subscription-like model that allows a person or business to register that domain name for a certain period of time (usually in annual increments). At the end of the registration period, you (the domain name registrant) have the option to renew the domain name registration for an additional period of time, or let it expire. It is important to renew your domain name registration, or you may find that all of your work building traffic and views to your site ends up benefiting someone else (in the instance that someone else registers your domain name after you let your registration expire), and potentially costing you more in rebranding.