''Box board'' is thin and smooth one one side, the smooth side can be used to print on or is used to glue foil onto. ''Graphic cardboard'' is used for jigsaw puzzles and hard book covers. ''Cup board'' is thin and coated, it is used to make paper cups and plates. ''Corrugated cardboard'' is usually used to make boxes, and it is sometimes waxed if they are to carry fresh food. There are other types of cardboard but you only asked for four.
