It is the job of the Internet Corporation for Assigned Names and Numbers (ICANN) that manages various parts of the internet. However, the thing that most people are concerned with is ICANN’s management of domain names.
You will not be registering a domain directly with ICANN, though. There will be two steps in the process. ICANN delegate the responsibility of the management of TLDs to various registry operators. Most TLDs will have their operator, while some will have two. For example; it is Verisign that are responsible for .com and .net registration.
You will not be dealing directly with the registry operator. Instead, you will be purchasing a domain name through a registrar. ICANN has accredited these. There are various registrars out there each different offering prices, but as long as a domain is available, you can shop wherever you wish (Namecheap, Hover, Moniker, etc.).
When you register your domain name, a WHOIS record will be created. It is handed to the registry operator.
All this information is available to the public.