You need a Registrar in order to register a new domain name or manage an existing one.
For new domains: When a domain is requested by a potential customer, the Registrar will check to see if the domain is available. If the domain is available, customer contact and payment information is gathered by the Registrar, and a request is made by the Registrar, to the registry, to obtain that domain for a customer.
For existing domains: The customer can setup an account with a Registrar and transfer their domain from it's existing location (a hosting company or different Registrar) to the new Registrar. The customer can then control the domain (and where it points) as well as their email addresses (if the account includes email service).
How is a Registrar different from a Hosting Company?
A Registrar should not be confused with a Web Hosting Company. However, things get a bit confusing because there are many companies that offer both Registrar and a Web Hosting services. You can work with either type of company to manage your domain name (and email addresses) as long as they provide the service you want/need at a reasonable rate. For example, you can use a Registrar to manage your domain name and email addresses without using their hosting services.