Representative means a natural or legal person established in the Union who, designated by the controller or processor in writing pursuant to Article 27, represents the controller or processor concerning their respective obligations under this Regulation.
The use of representative applies to companies that lack an establishment in the European Union but still process personal data of residents in the European Union. These companies need a representative to comply with GDPR.
An example could be an American company without an establishment in the European Union that sells products online to citizens within the European Union. This company need to have a representative within the European Union to comply with the GDPR.