A registry is an organisation or company in charge of a domain extension. There are hundreds of registries, one for each country code domain extension plus many others for generic domain extensions. Country code registries are almost always governmental organisations that manage the TLD for and on behalf of the local government.
Registries that manage generic domain extensions are usually commercial entities that sell domains under their TLD in order to make a profit. For example Verisign is a commercial company responsible for .COM and .NET, the DNC is the New Zealand registry responsible for the .NZ domain extension.