In Avatar: The Last Airbender, Zuko's entire mission to capture the Avatar is an attempt to redeem his honor, which he lost during an Agni Kai against his own father, Fire Lord Ozai. The Agni Kai is a ritualized one-on-one Firebending duel used to settle disputes and is one of the Fire Nation's most important customs. Interestingly, Zuko is involved in every single Agni Kai in the series, as well as two attempted but ultimately thwarted Agni Kais. This is a key aspect of his character, defining how he sees his place in the world.
Zuko vs. Ozai
When he was just a child, Zuko sat in on a meeting in his father's War Chamber. When a general proposed a strategy involving sacrificing an entire division of troops, Zuko objected vehemently -- a grave insult. There was only one honorable resolution to Zuko disrespecting the general: Agni Kai.
However, as the meeting took place in his father's War Room, he had to face his own father, Fire Lord Ozai. Zuko prostrated himself before the Fire Lord, refusing to fight. To teach his son respect, Ozai disfigured his son and banished him. Zuko can only return if he finds the Avatar, something believed to be impossible. This starts the prince's journey but also reveals he cares for others, and that his quest to redeem his honor is really about gaining his father's acceptance.
Zuko vs. Zhao
When Zuko docks his ship for repairs, he and his Uncle Iroh are invited to share tea with Commander Zhao, who commands the Fire Nation's fleet. As they drink tea, Zuko's crew are interrogated and revealed that the Avatar has returned. Zhao insists he will take over the hunt for the Avatar and cruelly mocks Zuko, who responds by challenging him to an Agni Kai at sunset.
The two face off while Iroh watches. Zhao is clearly the better martial artist, countering every one of the prince's attacks and driving him back. However, Zuko manages to break the commander's stance. Standing over his defeated opponent, Zuko throws a fireball but intentionally misses. Zhao calls him a coward, but Zuko assures him the next time they fight, no such mercy will be shown. The prince turns his back to walk away, and Zhao attacks from behind. Iroh blocks the attack, then admonishes Zhao for such dishonorable conduct. Though exiled, Zuko is still more honorable.
Attempted Agni Kai: Zuko vs. Jee
Jee is a lieutenant serving under Zuko's command. After the prince claims his men's safety is less important than catching the Avatar, Jee insults Zuko in front of the crew. The prince wordlessly assumes a fighting stance and they square off. Iroh intervenes, preventing the duel.
Later, Iroh tells Jee the events that led to Zuko's banishment, including how the prince advocated against sacrificing troops needlessly. When the ship is caught in a storm a short while later, Zuko sees Aang flying overhead, but does not give chase, instead prioritizing the safety of his men, and thus revealing his true priorities.
Attempted Agni Kai: Zuko vs. Azula
When Azula takes over Ba Sing Se with the help of the Dai Li, she ambushes Iroh and Zuko. As the two try to escape, Azula corners her younger brother, who challenges her to an Agni Kai. But Azula has no interest in honor, and so declines the challenge, having her Dai Li agents subdue Zuko instead.
She imprisons him with Katara, but after Iroh and Aang break them out, Azula shows up again. She gives Zuko a chance to redeem his honor by siding with her against Aang. This is the chance Zuko has waited years for, and he seizes upon it. Afterward, he is welcomed back by his father and heralded as a hero, but lives every day feeling shame for his actions.
Zuko vs. Azula
During the return of Sozin's Comet, Zuko interrupts Azula's coronation ceremony and the two engage in their final Agni Kai. Since the comet strengthens Firebending, they are both the most powerful they have ever been. While Azula has always been stronger, she has slipped into madness. Meanwhile, Zuko atoned for his sins, made peace with the Avatar, and studied Firebending under the ancient dragons, Shaw and Ran.
Their fight is nothing short of epic. Walls of blue and orange flames collide. Zuko counters Azula's every move, finally knocking his sister to the ground. Azula prepares to shoot lightning, but she realizes she can't beat Zuko, so cheats, attacking Katara instead. Zuko jumps in front of the blast, taking the full force of her attack. Katara manages to defeat Azula, then heals Zuko. In this final duel, Zuko proves himself superior in both combat and compassion. His honor now beyond reproach, he is crowned as the next Fire Lord.