The Foot-and-hand-ball is a small Sports object found in the 'Entertainment' catalog in the base game. It can be kicked or thrown, hence the name, "Foot-and-hand". It builds Fun, and also increases the athletic skill.
The foot-and-hand-ball resembles the ball used in American and Canadian football. The name pokes fun at the fact that, in those sports, the ball is almost always carried or thrown, and is rarely kicked.