There are lots of different ways to play games on the top of your checkerboard step stool. You can play standard checkers or chess, of course. But you might want to try French checkers, where unkinged checkers can't jump other pieces. (It makes for a long but exciting game!) Or there's Giveaway checkers, in which you try to lose all your pieces in order to win.
In Fox and Hounds, one player must try to get their single checker past a row of four checkers that are controlled by the other player. (Even kids 5 or 6 can join the fun!) Here are directions on how to play it:
Put the hounds (four black checkers) on the black squares on a king's row and the fox (a red checker) on a black square on the opposite king's row. The goal for the hounds is to surround the fox so he cannot move. The fox's goal is to move past the hounds to the opposite king's row. None of the pieces can jump, but the fox is able to move forward or backward. The hounds can only move forward. You "win" when either the fox reaches the opposite king's row or the hounds corner the fox so he can't move.