The Joker's Father was the father of the Joker.
According to the Joker, his father was "a drinker and a fiend".
Joker claimed that his father gave him his cheek scars due to his being terrified when his father brutally stabbed his mother sadistically with a kitchen knife when she attempted to use to defend herself. It was unverifiable if that was true, as the Joker also gave an account of inflicting the cuts himself because he wanted to show his wife that he didn't care about the facial scars that were inflicted on her by loan sharks.
The Joker also remarked that an older man reminded him of his father, whom he hated, which indicated that at the very least, Joker had a poor relationship with his father.