You need a domain name so people can reference your website.  This is a very important decision.  What happens is you go to a domain registrar to buy your domain name.  Typically it costs about ten to twelve dollars a year.  You can pay for longer than a year at a reduced price but figure about ten to twelve dollars a year.  Usually the site where you register will remind you when your time is up to pay for another year.

To give you some ideas about what to name your site,  I would suggest to try to use keywords that represent your business.  These are words or a phrase that describe the site.  Make it easy to spell and easy to remember.  Ideally you want a dot com website name.  Also, when you register, there will be a way that you can see if your site name is available.  My point is to have more than one idea on your website name.

As far as what website to go to for domain registration, most hosting websites offer that service.  In fact, sometime they offer free domain registration providing you use their service for hosting.

I suggest you read their terms of service.  When you read their terms of service does it tell you how to cancel the service if you decide to go to another hosting company.  If you have more questions call that hosting company and ask the questions that will help you understand their policy.

As far as a list of domain name registrar, you can go to Wikipedia.org or your favorite search engine and search for a list of domain name registrar websites.

Remember, go to  Youtube.com and search for how to register your domain name.  It is amazing what you can learn when you use YouTube as a source for learning.