The domain registration process is quite complex, and it involves three parties - the Registrant(which is usually the person or organization purchasing the domain), the Domain Registrar, and of course, the Domain Registry.
The domain registry is usually involved and responsible for the general administration of top-level domains (TLD). Some domain registries have control over second-level domains as well. A domain registry comes up with the domain-related policies for their TLDs, the domain's zone files, and registration requirements. They handle disputes between different organizations if the TLD they manage is involved. Typically, each domain registry is responsible for a single TLD, however, some of them can delegate more than one, such as Verisign, that caters registrations for .com and .net TLDs.
In addition to above-mentioned in regards to domain registries, some of them can manage third-level domain names, such as (co.uk, com.au, etc.) Usually, these are the registries managing ccTLDs(country-code top-level domains).