In American English, we only use the words zucchini and squash. Never courgette, and marrow to us means the stuff inside bones. But in the U.K., they can at times all mean the same thing, or very close.
For Americans, zucchini are a very specific type of summer squash that is dark green. Squash can refer to any summer squash as well as any winter squash, such as pumpkin, butternut, acorn, etc.