A website name is called a domain name. You must do a domain search first to make sure no one has bought that domain name already. Once you search for the domain you can find one that is right for you and purchase it. After that you need a hosting account. Once you have purchased a hosting account you can start building web pages. You can build one using code or an open source management system. Below I have listed a great place to search for website domain names.
Creating and Registering a Strong Domain Name Promo Code: IRDOM001AF Ad Code: AT NetworkSolutions.com