Mini-continent, with continental shelves and all? Volcanic upthrust? Part of Australia? Something unique?
New Zealand is an island arc, a landmass including a string of volcanoes along the edge of the Australian plate where the Pacific plate is being subducted underneath. The volcanoes have formed because the edge of the Pacific plate that is plunging underneath is melting to form magma pools, and hot magma will eventually rise back toward the surface. The Japanese islands are also part of an island arc, as are the Aleutian islands in Alaska.
New Zealand and Australia both lie within the Australian plate, but the bits that make up the continental parts of the plate have their own distinct geologic histories. This is possible because the plates themselves have been modified repeatedly over time - tearing off pieces and welding new pieces on.
Some useful links:
Map of the world’s tectonic plates
Diagram illustrating some of the terms and features associated with plate tectonics