Getting a domain name involves registering the name you want with an organisation called ICANN through a business called a domain name registrar.
ICANN (The Internet Corporation for Assigned Names and Numbers) is an internationally organized, non-profit corporation that has responsibility for all things Internet... including domain names etc.
To register your domain, visit the website of your preferred domain name registrar, check to see if the domain name (or names) are available. If they are, you will be required to pay a registration fee.
Depending on the type of domain name you register (eg; .com, .net, .com.au and so on), you would normally register your domain for one or two years, which you can renew every cycle to retain 'ownership' of that domain.
There are a thousands of domain name registrars. To find one simply type 'register a domain' into your favourite internet search engine.