When football was in its early years, the ball was made from animal bladders (usually pig). This is where the name pigskin comes from. The bladder would be pumped full of air creating a round ball that would be used in games. The ball would have to be pumped up often, as the stems originally created were not the most practical.
After some time, the bladder eventually was covered in leather, making it easier to grip and throw. The bladder itself was also eventually done away with, because the shape it created was not ideal for throwing. It was replaced with rubber that could be formed into more of an oval shape with points. The stem to pump up the ball was replaced by one that was inset into the leather, allowing for more practical pumping and throwing.
Despite the change in materials, the term pigskin is still used today. It can refer to the actual ball or to the game of football. A common phrase is to "throw the pigskin".
Professional NFL footballs are now made with vulcanized rubber and covered in a cowhide, giving it its iconic pointed shape and easily-gripped surface.