This is a slang term for the football, the actual game ball used to play the game of football. Before the modern era of football, the ball used in the game used was made of all natural materials. The outside of the football was, and is still, made of leather. The inside of the ball used to be made of a pig's bladder, which would be inflated to help keep the shape of the football. Currently, footballs are made by engineers with exacting specifications using man-made materials such plastics. The outside of an official National Football League football is made of leather.
This term could also be used to refer to the actual game, for example, "Hey Billy, do you want to play some pigskin?" In this case, the name of the ball is used, rather than the name of the game. This usage can still be heard today, and in fact, sports broadcasters sometimes use the word "pigskin" interchangeably with "football" to refer to the ball during games.