Full definition
Data controller is defined as: The data controller is the organisation that decides why and how personal data will be processed - the party accountable to regulators and to individuals under GDPR.
In a supplier relationship you are usually the controller. You decide what personal data is collected, how long it is kept and how it is used. The supplier that processes it on your instructions is the processor.
Two organisations can be joint controllers where they genuinely share the decisions, in which case GDPR requires them to publish an arrangement setting out their respective responsibilities.