House Lannister has ruled Westeros with an iron fist since the death of Robert Baratheon, but their grip has begun to weaken. Currently ruled by Cersei, who took the throne as queen regnant following the death of her children and father, the house has lost some of its key members to death and betrayals. Cersei reigns, with the support of her twin brother and on-and-off lover, Jaime, and while their army strikes a victory by wiping out House Tyrell, they are almost immediately obliterated by a surprise attack by the Targaryen army.
The three Lannister siblings have long been divided, usually with Tyrion on one side and Jaime and Cersei on the other. Tyrion is currently the hand of the queen to Daenerys Targaryen, to whom he has sworn fealty. After witnessing the deaths of his children and Cersei's betrayal of the agreement to ally with the North against the White Walkers, Jaime is disillusioned and departs for the North, presumably to ally with House Stark.