You're quite right there, dear. You see, when the master lost his mother, and his cruel father took that sweet innocent lad and twisted him up to become just like him, we did nothing.
~ Mrs. Potts explaining to Belle about the servants' regret over the King's behavior towards the Beast.
The King is the overarching antagonist of the 2017 Disney live-action film Beauty and the Beast based on the 1991 animated film of the same name. He is the late abusive father of the Prince (who was transformed into the Beast) who used to rule an unnamed French kingdom with an iron fist.
According to Mrs. Potts (the head of the castle kitchen), the King was known to rule his kingdom with his wife (the Queen) and young son (the Prince) by his side. Unlike the Queen (who is kind-hearted and caring to her son and subjects), the King is very arrogant and cruel in nature. He only appeared briefly when the young Prince stood by singing at the Queen's deathbed as she died from an unknown illness, right before he leads his son away from the scene. The King then took the opportunity to harden his son's heart through abuse to make him a vain and selfish ruler like himself. Of course, the castle servants secretly disapprove of the King's treatment towards the Prince, but they were too scared to do anything about it as they feared of any retaliation that the King would inflict on them.
The King's horrible treatment is what caused the Prince to develop an arrogant personality towards his own subjects during his young adult years, even raising taxes on the local villages to fund parties for his own amusement. What ultimately became of the King is unknown, but he most likely died, and the Prince took over; in any case, the Prince followed in his father's footsteps. It was not until one winter night during a debutante ball, a visiting enchantress named Agathe, disguised as an old beggar woman, placed a spell on the Prince as punishment for denying her any shelter, transforming him into the Beast and the servants into household objects. After erasing the memory of the castle from the townsfolk, Agathe tells the Beast that he can break the spell by learning to love another and earn her love in return before the enchanted rose's last petal falls, otherwise his monstrous form will remain and that the servants will become permanent antiques, much to their distraught.
Because of this event, the Beast felt so extremely ashamed of his actions and completely despised his cruel father for raising him to become a selfish tyrant; even furiously tearing up a family portrait of himself and the King in retaliation (though the portrait of the Queen remains intact as the Beast still mourns for her death, and couldn't bear to ruin his mother's portrait due to his genuine love for her). Even the servants felt depressed over their role in the event, as they are full of regret of letting the Beast becoming so arrogant due to the King's horrible treatment. Eventually, a young village bookworm named Belle reached out to the Beast's heart after he saved her from a wolf pack. To that end, Belle negates the King's treatment by helping the Beast unleash his inner humanity before giving out her love for him. This effectively breaks the spell following Gaston LeGume's death, turning the Beast and his servants back to normal, much to their delight and bringing the King's dreadful legacy to an end for good.
The King's personality and behavior reflect that of Gaston LeGume (the main antagonist of the film):
They are both selfish leaders.
They tend to abuse those who are close to them (The King to the Beast, the Queen and the castle servants; Gaston to LeFou and his buddies).
They sense and seek an opportunity for glory and power by any means necessary (The King hardened the Beast's heart to become an arrogant ruler following the Queen's death; Gaston goaded the villagers to hunt down the Beast in an attempt to have Belle for himself).
However, while the 2017 Gaston is implied to have a severe PTSD, the king had no Freudian excuse.
Given the King's unconcerned behavior towards the dead Queen as he leads the Prince away from the deathbed, it might be possible that he may have neglected her illness in hopes of driving their son to become an arrogant ruler like himself (this kind of attitude giving him resemblance to King Stefan's attitude in which the latter neglected his wife's death due to his obsession to hunt down Maleficent).
It is also possible that the King may have actually poisoned his own wife as a means of removing her influence on their son, like how it is theorized how Lady Tremaine murdered her husband so that she can fully abuse her adopted daughter.
The King is responsible for the events of the film, as his abusive treatment towards the Prince is a clear explanation why the Prince became so selfish and arrogant that it incited the enchantress to set off the film's plot in the first place.