You don't actually own your domain name. You rent it. You pay an annual registration fee for the exclusive right to the domain name for a period of 12 months. You need to renew the registration each year in order to maintain your right to use the domain name.
Domain names are registered through a central Internet registry which involves things like Doman Name Service (DNS), DNS servers, and DNS configuration. None of this is difficult IF YOU KNOW HOW but it can be daunting to the beginner. There are a number of certified domain registration agents where you can register your domain name. Many of these will provide help with the process but expect to be "up sold". Don't let them convince you to buy things you don't need!
Domain names can cost anywhere from a few dollars up to $100.00 or more depending on where you register them and what is bundled in the fee. A domain name by itself shouldn't cost you more than about $12.00 to register. The are available for even less, and sometimes free, especially if they are included with website hosting or other services. Be sure you know what you are getting.
Some people register domain names on your behalf, and let you use them, but they actually register them in their own name. Sometimes this is okay because it provides for easier handling by the person doing this. Sometimes it's not a good idea, especially if you cannot take the domain name to a different registration service if you don't like the service you are getting. Ask and be sure you understand what the registrar is doing.
If you want your own domain name, but are concerned about being able to do it yourself, the Wiz (Steve Henry), known for his help with all things online, will handle the process for you for $45.00. This price includes the domain registration fee, configuration of DNS, and any changes or moves you require during the 12 months the registration is in effect. He can be reached at thewiz@ourhutch.com.