The word paternal describes anything that has to do with fatherhood, like your dad's paternal habit of giving you advice, always including some "when I was a boy..." tales.
Paternal originates from the Old French word of the same spelling, meaning "of a father." For example, your paternal grandparents are your father's parents. (Your mom's parents are your maternal grandparents.) Paternal describes your father specifically, or fatherhood in general, like common paternal activities like father-son fishing trips and making sure the guy who is dating his daughter is respectful and sincere.