Domain name registration is the process of registering a domain name of your choice. This typically happens after you have purchased a domain ( .COM, .IN, .NET, etc. ) from a registry or a registrar.
ICANN is the internet body that governs the domain and registration space. When it releases domains, it announces the phases in which the availability of the domain will take place. Sunrise period, General availability period etc. I can talk about this in detail but I am trying not to digress from your question.
Since you touched upon the topic of domain name registration, let me also tell you that you can choose any domain that you want. It is nothing but the the name of your website, in common usage, like a family name. Once you search for that domain name, the server will function to provide you the files of that website.
It is a name of your website through which user access your site.
If you are looking to register a domain and a domain name, Many Provider today giving you these services. However, I personally prefer Nixuz
since they provide easy domain registration services.