Custom Resource
Custom resources are extensions of the Kubernetes API.
They are not available in a default Kubernetes installation and have to be installed/created separately.
All requests are coming to the API server as usual.
Custom Resource Definition (CRD)
CRD is a way to define a custom resource.
To work with CRD, you need to create a Custom Controller to handle CRD requests, because the API server doesn't know how your CRD should behave in different situations.
Controller vs Operator
Controller works on vanilla Kubernetes objects.
Operator is a controller that works with custom resources.