To me they all mean different things. Harbor is the general area, mainly the body of water meant for anchoring ships, while port mainly refers to the general vicinity where the docks are located, and the docks themselves refer to the physical structure which boats dock themselves to for purposes of transfering people and objects. (American English speaker, here)
A port is a location on a coast or shore containing one or more harbors where ships can dock and transfer people or cargo to or from land. Port locations are selected to optimize access to land and navigable water, for commercial demand, and for shelter from wind and waves.