Pig Chance is a characteristic that becomes available after purchasing the Extreme Gardening board. It (as the name implies) increases the chance that the animal dropped will be a pig. Pig change is increased primarily by growing carrots. The pig chance meter is located in the lower left hand corner of the screen.
Pigs don't appear to do much, but they have different effects depending on the board:
Extreme Brickout, Starshooter, and Lawnmowing: when a pig falls off the screen, it yields $5. Comparatively, when a sheep (or whatever it is) falls off the screen, it yields $1.
Extreme Animal Jumping: I have not noticed a difference between pigs and any other animal.
To increase your pig chance for a while, go to the garden board, then grow and harvest exclusively carrots for about a minute. You can watch your pig chance increase in the sidebar on the lower left hand corner.