The characters in Avatar: The Last Airbender are all incredibly complex and deep, their personalities are visible from day one. They have their own quirks, their own weaknesses, and their own strengths.
Each member of the core group brings something to it that the others don't, and the same goes for the villains. Each person brings their own element to the show that is needed. The same goes for battles, too. With so much fighting going on in the series, it's very easy to see which characters are bringing what to the table.
Sokka: His Leadership
At one point during the show, the entire cast had to work to figure out what it was that Sokka actually brings to the group, but it is without a doubt his leadership. During the Invasion of the Fire Nation, Sokka is a key figure in the army's victories, leading the charge as a commander. He also works every day to keep the group together. Sokka is the glue that holds everyone together. Without him, who knows if the Fire Lord would have ever been taken down. Things would have been very different without Sokka either way.
Toph: Her Raw Power
Toph might be blind, but she's easily the most talented earthbender in the entire series. She has a deep connection with the ground that allows her movements to be incredibly precise in a way that very few are able to master. She's an amazing bender, through and through.
This translates into pure, unadulterated power. Toph is simply overflowing with it. She even created a brand new form of bending, who else can say that? Toph was key in Avatar Aang beating the Fire Lord, as without her as a teacher, his grasp on earthbending would have never been as strong as it ended up being.
Katara: Her Intellect
As the sole bender of the Southern Water Tribe, Katara had quite a lot to prove throughout her journey, and she did so handily. Katara was the perfect first teacher for Aang, as she was able to teach him the basics of bending very quickly before they learned more advanced tactics together. While Katara has flown off the handle a few times in the past, she's generally very smart when it comes to keeping her emotions in check and bending with only her intellect. Katara is smart, there's no denying that. She knows when to fight and how to fight depending on the situation that she's in.
Aang: He's The Avatar
Obviously, Aang's biggest asset is that he's the Avatar. He is the only person in the world that is capable of mastering all four elements. And Aang is naturally gifted when it comes to that, quickly reaching Katara's level of bending early on into his training. Another important factor of this is that Aang is young, easily the youngest Avatar who's ever had to do anything of note in their life. Aang manages to beat the Fire Lord while he's still a child, which is a momentous feat that very few would be able to accomplish even as adults.
Zuko: His Growth
At the beginning of the series, Zuko is brash, thinks little about his actions, and is furious at the world around him all of the time. However, little by little, episode by episode, he grows. He begins to get a handle on his emotions, and his bending.
He slowly realizes that his anger is directed at someone it shouldn't be, Aang. And before long, he's training the Avatar in firebending. There's no better teacher for him, either. Zuko manages to even control his emotions enough to redirect lightning, which shows just how much he grows throughout the series.
Azula: Her Raw Power
Just like Toph, Azula's greatest asset is her raw power. Where Azula collapses is the lack of control that she has over her emotions, which, in part, along with a psychological break, is what causes her to lose control in her final battle with Zuko and Katara. Every time that we see her bend before her final encounter, however, it is pristine. She is incredibly adept in her bending, so much that it's frightening. If it wasn't for the break she had near the end of the show, she might have just taken down Zuko and Katara.
Iroh: Emotion
It's inarguable that Iroh's greatest strength is his emotion. He's in touch with his inner self in a way that no one else is. This is a key factor in Zuko's arc throughout the series. If he didn't have Iron at his side, constantly challenging him to look inward and grow as a person, he might not have ever been able to reach the heights that he eventually did. In that sense, Iroh is one of the biggest factors in Aang's eventual win over the Fire Lord. Iroh cares about Zuko more than anything, and it is his teachings and love that help him turn into the man, and the Fire Lord, that he becomes. Zuko needed Iroh more than he could have ever known, and Iroh never gave up on him.
Suki: Her Bravery
Despite being a non-bender, Suki is an incredible fighter who was a huge asset to Team Avatar as they sought to end the 100 Year War. While she did lose to Azula's crew at one point, she still managed to be a huge ally and press on, becoming a key figure on the final day of the battle. She never lost hope, despite what she had to go through.
Mai: Her Intelligence
While Mai was an ally to Azula for some time, she knew eventually that it was time to quit. She wasn't going to become a blind follower that completed any task that Azula put in front of her. She was very much her own person that thought through every decision that she made rationally. Not to mention those daggers of hers were just too fierce.
Ty Lee: Hand-to-Hand Combat
Ty Lee's hand-to-hand combat skills are unmatched. She's an incredibly adept and nimble fighter, and also has the ability to chi block, which stops a bender in their tracks and paralyzes the victim for a very short time. She utilized this often throughout her time on the show, even doing so to save Mai from Azula's rage after her betrayal.