I'll get (PHP) Laravel-framework specific in my response and respond from that specific architectural view point.

"Controller" is what stands between the incoming request to your application and the business-logic operations it performs, while "Service" is what wraps around that same business-logic opeations and is called my the controller.

The following flow might be helpful in wrapping one's head around the concept:
client sends a request -> controller (validation, calling the necessary service to perform business logic and passing the request to it) -> service (computation, db checks, resource mutation, etc) -> response back to controller (formatting, view) -> client receives response.