Olivia Monroe is the LAPD Chief of Police. She had been a homicide Lieutenant with great ambitions when she met Lucifer who saw that in her and used it for his own aims, making a deal to help make her Chief of Police, in exchange for his being named a civilian consultant for the LAPD.
Olivia first appeared during the conclusion of "Manly Whatnots" attending the crime scene where Chloe had given into Lucifer's urging and had actually shot him, then told him it could cost her her badge, but Lucifer apparently told Olivia that if Chloe "hadn't acted in the way that [she] did, then, well, all hell would've broken loose." Which while technically true, was not the whole truth. Olivia tells Lucifer, "I'm sorry that you got caught in the crossfire, Mr. Morningstar," and she then clearly is effected by his presence in a way that makes Chloe roll her eyes.
She returns in "Sweet Kicks" when Lucifer tells her that he wants to take part in the new investigation Chloe is on. Olivia tells him that although she realizes he has been helpful in the past, his thrill seeking approach isn't working for her, "exploring mortality is not really what I do here."
Lucifer then notes that she is an ambitious one, "brimming with desire," and asks what her goal is. She responds, "If I'm going to be the first female chief of police, I need a big close." Lucifer tells her he can get her all the support she needs, but she has to allow him a role in the investigation.
With the successful wrap of the case, Olivia makes an appearance on the news and Chloe tells Lucifer that Olivia will be police chief because of him. He tells her, "Olivia got what she wanted, and I got what I wanted....I am now an official civilian consultant for the LAPD "