The Joker's Father was the father of the Joker.
According to the Joker, his father was "a drinker and a fiend".
He 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 she attempted to use to defend himself. It is unverifiable if this is 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 inflicted on her by loan sharks.
The Joker also remarked that an older man reminded him of his father, whom he hated, indicating that at the very least, he did indeed have a poor relationship with his father.