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.
NOTE: Networksolutions is great for figuring out what domain names are available, but I recommend using Godaddy for the actual registration. 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.
The ".com" extension is most often used for business websites but there are other acceptable extensions like ".biz" and ".net". You may want to pick a domain name where the ".com", ".net", and ".org" extensions are all available. There is an cost for each domain extesion, but you may want 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. If it is not a business website, there are other less expensive extensions like ".info".
Many hosting services offer to register your domain name, but I do not recommend doing that. 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.
For an extra fee you can keep your registration information private to protect your name, email address, and phone number from being made public.
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.