Generally - the natural occurrence of facial hair, also known as a beard, which tends to grow on a man's face, if he chooses not to shave for an extended period of time.
Specifically - Jake Gyllenhaal's beard.
The term face pet was first coined by "lemniscate", on September 29th 2010, when posting on the popular Jake Gyllenhaal fan blog called Wet Dark And Wild.
The word combination face pet was engendered in reference to:
1) the tendency men have to subconsciously stroke and touch their face and/or facial hair; often a form of auto-erotic touching when flirting to indicate they're interested
2) the health benefits discovered by researchers at the University of Warwick with regards to stroking or playing with one's pet
Jake gently strokes his face pet whilst staring into the distance, lost in a wistful gaze.
to stroke someones face in order to cool their fears and soothe their mood
As Clyde faded out of consciousness in the arms of Mary we all knew that the face pet she was applying would have no effect in this dire situation.