Only 16 years old and already he's been banished from his home in the Fire Nation. At the show's beginning he has been sailing around the world with his Uncle Iroh and a crew of firebenders for two years, in hopes to find and capture the Avatar. Doing so would restore his honor in his father's eyes and give him back his birthright as heir to the Fire Nation throne. Because of this, Prince Zuko is very determined to capture Aang. He is learning firebending from his Uncle Iroh, and is a worthy opponent.
The reason for Zuko's banishment is directly connected to how he got the large scar on his face. When he was 13, Zuko was allowed to attend a war meeting in his father's war chamber. But when he spoke out in defense of a newly recruited battalion that was suggested by a general to be sacrificed as bait for the enemy, his father took the outburst as utter disrespect. The only way to resolve the issue was by an Agni Kai; however because the outburst was in the Fire Lord's war room it was he whom Zuko disrespected, and who he would have to duel.
Yet Zuko, being the loyal son that he is, refused to fight and begged for mercy. Fire Lord Ozai wouldn't hear it though; he said that Zuko would learn respect, and Suffering would be his teacher. (yeah, Ozai is SO going to win the Father of the Year award. -_-') The burn on Zuko's face is the result of their one-sided duel. In addition to being mutilated in front of an audience of hundreds including family, soldiers, and civilians; because Zuko refused to fight him, his father called it a shameful weakness and banished him from the kingdom. The only way he would be accepted back with honor was if he captured the Avatar. Who, at that time, had been missing for a hundred years. Ouch.
However.... Despite this unjustly harsh treatment, Prince Zuko is eager to prove himself to his father, and to some level, honestly believes that he will be accepted back. While it's clear that Zuko idolized his father when he was younger, it's unknown how much of this attitude remains. During a forced conversation with Commander Zhao in ep3, Zuko calls his father a fool for believing that the other tribes would willingly follow him. Not to mention how his time away from home has changed him. And yet, his obsession with getting the Avatar is proof enough that he is still starving for his father's love.
The only person who keeps Zuko from totally losing it is his uncle, Iroh. Iroh has always been looking out for Zuko, even before he was banished. Though Zuko may not always seem respectful of his uncle's support, you know that deep down he does appreciate it, and in the later episodes you can see signs of that. Iroh's wise and laid-back nature balances out Zuko's impatience and rage, making them a great duo and leading to many comedic scenes. ^-^
Prince Zuko is hotheaded, impatient, short-tempered, and determined in everything he does. He's also honorable, sees no need for unnecessary violence and killing (as long as he gets what he wants), and has daddy issues. And while he's obsessed with capturing the Avatar, he's slowly realizing that there are other things more important, like the safety of his family and/or crew. He often meditates alone in his room in front of a large dragon motif; being taught by Iroh that a level head is a sign of a good leader. He has issues with Commander Zhao and clearly dislikes him, although its unclear why (you know, besides their total clash of personalities). Zhao constantly refers to Zuko as a traitor and even questions his loyalty to the Fire Nation, but what caused such remarks is also unknown. In the few flashbacks we see of Zuko, it's clear that he cares for the Fire Nation and its people.