And in doing so, Zuko does the right thing, finds fulfillment, is enlightened ("I understand"), is redeemed, earns Uncle's forgiveness, and attains personal happiness.
So where does Mai fit into all this? Well, who is Mai?
Mai is a nobleman's daughter. Her family is clearly of immense status. They wer gifted the city of Omashu, live next door to the Royal Palace, she grew up playing with the Royal children and attended the same school as Azula. Mai is deeply bored with palace life and repeatedly says as much.
She lacks empathy for people, Fire Nation and foreign. Her concept of an amusing evening is sitting recumbent on her couch while her prince boyfriend screams at servants to fetch her ornately decorated fruit tarts. So charming is this idea to her that she eagerly suggests it to Zuko as a way to cheer him up when he's worried about not getting into the war room (you know, that thing his career and physical safety is dependent upon). Mai is the kind of rich brat so self absorbed and oblivious she legitimately sees nothing wrong with bossing poor people around for a laugh.
And if extends to the peasantry of the Earth Kingdom of which her father rules. The moment she sees Aang she leaps at the opportunity to hunt them down. This is before Azula's arrival. Unlike Ty Lee, a genuinely gentle hearted person, who must be coerced into combat under threat by Azula, Mai eagerly jumps at the chance to throw knives at Earth Benders.
She's also a whiny liar. Mai says that her parents gave her whatever she wanted, repressed her, and forced her to sit still and quiet throughout her childhood to maintain her father's career.