Once you decided to set up a web site for your personal use or business purpose, you have to find identify a domain name for your web site through which you are known and accessed by people browsing internet. The domain name system developed in 1980 and a domain name has got two parts one is either your name or company name and second part separated by a dot such as .com, .org, .ae, .sg, .in, .gov, .ed, etc. Usually commercial establishments uses the name ending with .com, for organizations such a wikipedia, it is ending with .org, .edu for universities or educational institutions and for country names such as .ca for Canada, .jp for Japan, .in for India, etc.
The domain names are registered with ICANN (Internet Corporation for Assigned Names and Numbers) by registered members and domain names given as per rules and procedures for DNS (Domain Name System). Many people believe that URL (Uniform Resource Locators) is domain name and in fact both are different. A URL will be like http://en.wikipedia.org/wiki/abcdfd and the domain in this URL is only the name Wikipedia.org. Also, domain name is not same as IP address and many people confuse with both the terms. Most American Servers use Top Level Domain (TLD) names with three letters such as .com or .edu and countries other than America uses TLD in two letters such as .ca, .au, etc.
When you find the right web host for your web site hosting, you can check with them the availability of domain names and you have to keep best suitable names selected before you approach them. You can check availability of the domain names in the web site of most of hosting companies. Once you have selected the domain name it can be registered.
Domain names cannot be registered twice means similar name cannot be registered. ICANN have the control of distributing the names. Many hosting companies offering free domain name along with other free offers. If they are offering free, you should ensure that the domain name is registered in your name and not with the host company.