A Registrar is an organization in charge of registering who "owns" what Domain Name. There's more to it than that, but that's a basic explanation of a Registrar - they help people register Domains.
So, who's in charge of the Registrars then?
The short answer is ICANN or, "Internet Corporation for Assigned Names and Numbers" and, according to their website:
What Does ICANN Do?
To reach another person on the Internet you have to type an address into your computer -- a name or a number. That address must be unique so computers know where to find each other. ICANN coordinates these unique identifiers across the world. Without that coordination, we wouldn't have one global Internet."
ICANN basically runs the Domain Name System as we know it, and sets the rules and policies for Registrars and Domain owners to follow.
There's a lot to it, so feel free to contact us if you need help understanding anything about Domains or Registrars.