Before you can use a website name you must determine if anyone else has already registered (owns) that name. This can be done by doing a name search at a domain name registration website. I use http://www.networksolutions.com because it allows you to enter multiple names and shows possible extensions. DO NOT register the name yet as that will be free if you continue to follow this procedure. Domain registration services generally charge a fee of from about $10-$35 per year to register a domain name. Domain names are generally registered for from one to ten years at a time and the registration must then be renewed. If a domain name is not renewed that domain name becomes available to others.
For business websites it is best to use a ".com" extension. If you can, pick a domain name where the ".com", ".net", and ".org" extensions are all available. It is a good idea to also register all of these extensions to prevent anyone else from using them. Someone might use them for an undesirable type website and people might go to it thinking it is your website. You can register other extensions for your domain name after you have completed the host sign up procedure. I recommend Godaddy. It is very inexpensive to register several extensions and set them up so they will forward to your primary extension. If it is not a business website there are other extensions that are suitable.
Many hosting services offer to register your domain name for free if you purchase their hosting service. While you will save a little money that way, it is not recommended. If you keep using that hosting service it is not a problem, but if you have problems with your hosting service, it happens, it is realatively easy to change hosts but moving your domain registration can be a real pain! That is why I recommend using a separate domain name service like Godaddy.
NOTE: Some domain name extensions are country unique. The above registration method works for most domain name extensions. If you are not able to use this method to register a particular country-unique domain name extension you will have to do a browser search to find an organization that registers the desired extension.