It has been two years since Aang defeated the Fire Lord, since Azula was sent to an asylum, and since Zuko was crowned Lord of the Fire Nation. In these years, the world has more or less been peaceful. The band of friends that made this possible split up. They're still in contact, but are no longer traveling with each other.
Aang became entwined in political issues straight after the war. He now travels mostly between the Fire Nation and the Earth Kingdom, helping with reconstruction and government whenever he is needed. Ever since the end of the war, the lingering question of what the next air cycle will bring to the Avatars' spirit has haunted him. He wonders if the war wasn't his only priority as Avatar.
Katara returned to the South Pole. For a while, at least. She resumed her role in cleaning, cooking, etc. for the tribe. She felt troubled, though, that after all she had been through she could settle for the monotony of life in the South Pole. Worse yet, Aang nearly never visited her. She would consider starting to, well, see others, but most of them were too old or too young for her taste. To make matters worse, Khana died just a year after Katara's return. She felt like she was settling for monotony, felt shallow for not feeling there wasn't anyone that she could be in a relationship with, and had no family save her father anywhere near her. She soon found, though, that there were others in the Tribe who felt the same, whether they had little family, no interest in the South Pole's pick of men, or felt as if they were in a rut. They all became friends quickly, and decided to seek out a better future in the North Pole. Two days before our story begins, Master Pakku dies. Katara was helping him teach some of his students, and now the responsibility is hers.
Toph went back to her parents. They got much needed closure, and her parents decided to let her have more freedom. She still had to keep a few secrets to be able to continue her 'career', of fighting. Her parents do, however, expose her to the general public, allow her to have visitors, and allow her to give earthbending lessons. Despite being one of, if not the best, greatest earthbenders in the world, she feels there is something missing from her technique. She wishes to find whatever this is.
Sokka and Suki have stayed together. Shortly after returning to the North Pole, Sokka set out for Kyoshi Island to see Suki. When he got there, he found that the Kyoshi Warriors had begun to allow men into the group. Sokka became one of them. After the war, the Kyoshi Warriors became renowned for their work. Soon, people began to ask the Kyoshi Warriors to help them with problems in their own towns. People came to train with the Kyoshi Warriors, and left to train others. The Earth King soon decided the original Kyoshi Warriors, along with some newer members, would replace the Dai Lee as guards and advisors. Now, Sokka and Suki, along with the others, live in one of the innermost rings of Ba Sing Se. Sokka and Suki are now married, and Suki is two months pregnant.
Zuko hasn't had as easy a ride as some of the others, though. Right after the war, he had to plunge into the reconstruction of the Fire Nation. The educational system needed to be liberated; laws needed to be fixed and destroyed, free thought struggled to break through 100 years of corruption. It seems as though the Fire Nation is slowly coming out of the aftereffects of the Phoenix King's reign. But there are other problems. Some people think Zuko should follow in his father's footsteps and try taking over the world again. Also, nearly a year after the war, Mai, 7 months pregnant, died in a horrible accident. Ursula found her way back to her old home the day of the funeral. It turned out that she needed months of time to scrape up enough money to get to the Fire Nation. She had been living in the poverty-ridden outermost ring of Ba Sing Se in the years she had been gone. Now Zuko is being pressured to find a wife to bear him an heir, must reconcile his mother, and soon he will have to deal with a rising rebellion that feels he isn't as good a leader as the 'true' Fire Lord was.