You should always own your domain name; ensure whoever acquires it for you actually registers it in your busines name. This way you are not locked to any one website service provider in the future.
Domain names are registered at what are known as "Domain registrars". Once registered, a domain name is yours to use as long as you keep paying for its renewal. You pay the registrar directly for initial registration and ongoing renewals (every 1-2 years). Different registrars charge different prices, though essentially the service is the same.
Having a good domain name is very important. We have a blog post written about this topic.