Zuko was Azula's faithful, courageous, and persistent yet underappreciated brother. Ozai saw Zuko as a failure and Azula had several conflicts with him because she was trying to impress her father.

Zuko and his mother Ursa.
Early Life
As a boy, Zuko often challenged Ozai's judgement. This led to him being verbally scolded. Ozai berated him as "soft" and "mentally weak". Ozai even considered killing Zuko when he thought he might not have been born a firebender, but Ursa prevented that from happening. From that moment, a bond between Ursa and Zuko was formed. She always looked after him and stood up for when. When Ursa was around, Ozai limited how much he intimidated his son.

A younger Zuko hunting Aang.
Hunt for the Avatar
Zuko was maimed and banished by his father for challenging a general's order that would have needlessly sacrificed the lives of many young FIre Nation recruits. Ozai saw the act as a move in defiance. Unlike Zuko, Azula never openly challenged Ozai. She still didn't like unnecessary bloodshed and respected Fire Nation soldiers more than her father. But Azula, was very careful in how she made decisions and ultimately would accept whatever decision she made. Even Ozai, who didn't always agree with Azulon, would accept his decisions when they came.
By Zuko making his own stance for the people, Ozai saw it as a personal challenge. The Firelord even considered forgiving his son after beating him into submission during an Agni Kai. However, Zuko was not willing to fight his own father and refused. Ozai was further infuriated and embarrassed by Zuko's act of kindness. He believed he needed to beat the compassion out of Zuko and make him into a war machine.
Ozai burned the left side of Zuko's face. Despite his best efforts, Zuko never lost his compassion. Ozai then stripped Zuko of his honor and would not allow him to stay at the palace until he captured Aang and brought him to Ozai. An exception may have been made if Zuko managed to "kill" the Avatar. Zuko would later meet his father after apparently "killing" Aang with the help of his sister. When this happened, Ozai welcomed him back home.

Joining Team Avatar
Zuko continued his attempted hunt of Aang after the Fire Nation's failed Siege of the North which was the first massive defeat they faced since the Siege of Ba Sing Se. Zuko mainly defected because he realized that everything he had with Ozai was fake and fragile. He also learned that Aang, Katara, Toph, Sokka, and the citizens of the other nations were not his enemies. He merely drew arms against them due to a conflict that was created for the acquisition of power.
The only people who really benefited from that acquisition were Fire Nation nobles. Even then, it came at a great moral cost and could always inspire revenge from the members of other nations. Zuko eventually confronted his father and told him the truth. Ozai attempted to kill him by using an advanced form of subbending that only a handful of firebender's possessed. Ozai fired a surge of lightning at Zuko but Zuko used a technique that Iroh taught him to redirect the lightning back at him. The move could have killed Ozai but Zuko instead only used it to humble him.
Abilities
Zuko was a deadly firebender. He was a step behind Kanna and Mai in agility. Zuko was faster than Katara, Macai, and Rosh. On the other hand, for some time Zuko's defense was lacking which left him exposed for counters and other attacks. As Zuko got older, his defense improved and this became more noticeable toward the final months of the war. Zuko's flames were larger than all firebenders except Ozai and Zhao. However, Zuko's skill level was well above Zhao's. Zuko became notably dangerous whenever he raged in combat. While raging might make many benders more vulnerable it actually made Zuko more passionate, had his attacks more unpredictable, and caused him to fire larger flames at his opponents at a faster rate than normal. Zuko was said to have more heart than anyone else in the world, he always put everything into his fighting.

Firelord
Against all odds, Zuko would eventually become the Firelord. As Firelord Zuko learned how to create sparkling rainbow fire. His rainbow fire was primarily light orange in color but had shades of every color within it. This made his flames hotter than most Firebender's but they were still a touch less hot than Macai's.
Mai
Zuko's love was always concentrated on a young woman named Mai. Unfortunately, they had a rollercoaster relationship that went up and down for many of their early years.