You probably know The Last Airbender's setup by heart, thanks to the fact that they repeat it at the top of every single episode, but let's refresh just in case. Katara and Sokka, siblings and members of the Water Tribe, found Aang encapsulated in a massive iceberg off the shore of their village. Over time, the group pieces together enough information to realize that Aang's been inside that iceberg for at least 100 years. He has no idea that everything went to shit when he disappeared, that the Fire Nation has taken over and destroyed much of what was in its path. More tragically, he has no idea that every last Airbender has been wiped out. Except him.
Aang eventually reveals that he's the Avatar, but only after he has to. When prodded by Katara, he explains that he never wanted to be the Avatar. He kind of just wanted to be a regular kid -- and for once, he was. No one knew who Aang was or had any expectations of him.
Despite being frozen for a century, Aang remains insistent that the damage the Fire Nation caused couldn't have been that bad, especially when it came to his people, the Airbenders. The only way to get to the Southern Temple is by riding a flying bison, and the Fire Nation certainly doesn't have those. When Aang sees that his home is indeed destroyed, he's upset, but still certain that the other Airbenders have escaped.
Though he's disheartened by the absence of his fellow Airbenders, the gravity of the situation didn't hit Aang immediately, in part because Katara and Sokka tried to shield him from the reality of his nation's genocide. It's not until he stumbles upon the bones of his late mentor Gyatso when everything becomes painfully clear.
Airbenders were wiped out long before Katara and Sokka were born -- the Air Nomad genocide is closer to ancient history for them. But for Aang, who's been trapped in ice for 100 years, it's a fresh, deep wound.
Zuko is a very angry dude. Granted, he was banished from the Fire Nation, but he's pretty whiny about it. His obsession with the Avatar seems slightly unfounded in the first half of The Last Airbender. He's willing to sacrifice anything (and anyone) to catch Aang. Zuko pushes everyone to the edge, including his own crew. It's in The Storm that we learn his irrational anger has a rational origin story.
In The Storm, unsurprisingly, a storm is coming, but Zuko is adamant that his crew should sail on to find the Avatar. Things get so heated -- literally, smoke is coming out of Zuko's hand -- that Zuko's uncle Iroh has to break up an erupting Agni Kai. That's a duel sort-of thing in the Fire Nation; we'll get back to it later.
Zuko storms off, as you'd expect him to. That leaves his crew to complain about their leader and his man-child behavior. And I don't blame them... it's apparent that Zuko would risk the lives of all his men to find the Avatar, and he admits as much aloud. Around the fire, Iroh reveals why Zuko is obsessed with the little Airbender.
We flash back to a scarless Zuko, intent on entering his father's Fire Nation war room. That's one of the perks when your dad is the Fire Lord. Zuko is allowed entry, but told to sit quietly and just listen, and for his part, he agrees. After all, Zuko is set to become the Fire Lord himself one day, and it makes sense that he wants to learn what goes on in the War Room.
One of the generals suggests sacrificing a group of young soldiers as a distraction, so that the rest of the army could attack unhindered. It's a horrific plan, one that angers Zuko to the core. And he's not quiet about it. Despite being told to not speak during the meeting, Zuko loudly opposes the plan, calling it an act of betrayal. Iroh points out that Zuko was right -- the general's plan was cruel. However, Zuko's outburst was considered far worse by those in the room.
Speaking out of turn like this is considered terribly disrespectful, and only settled by an Agni Kai. Zuko accepts, not knowing that he'd actually have to duel his father.
Once he realizes his error, Zuko refuses to fight. Begging for mercy, Zuko tells his father that he only spoke out because he wanted what was best for the Fire Nation. This only makes his father more angry. He wants Zuko to defend his honor -- and when he won't, the Fire Lord strikes his own son in the face while he kneels before him, crying.
Because Zuko wouldn't fight, his father not only made him suffer physically, but banished him from the Fire Nation. That is, until he's able to capture the Avatar -- who, at that point, had been missing for a very, very long time. It's sort of like exiling someone until they can bring you a tall leprechaun.
Up until this point, we've seen just one side of Zuko. The side that's merciless and unforgiving. This is what his father instilled in him. Through the flashbacks, though, we see what he used to be -- and we see that capturing the Avatar is the only thing that brings him hope in getting that life back. Not a lot you can do about that permanent black eye, though.