The rare non-polygonal engines use other methods to create objects, like voxels . With a triangle-only engine, you can still use polygons with more than three sides, but the engine always sees them as tris . However, if you are able to use quads, the engine is then able to store them as only four vertices apiece, instead of the six that would be required to store the two triangles that make up the square shape. Engines that support quads usually support plain old triangles as well, for those difficult shapes where quads won't work. Quads save on the size of the geometry file, which saves memory , and it saves on the amount of calculations the engine has to do to transform the quads before rendering, which can ultimately speed up your frame rate .
Every little bit helps.
The same holds true for n-gons, except they can save even more vertices. The hexagon, for example, is stored as only six vertices, instead of the twelve verts needed for plain triangles to describe the same shape. There really isn't a specific term for this. Which really isn't all that often, as most engines do not support them. Each quad has their vertices in the same shape.