They are the generation that grew up on pain. From their earliest moments, to their dying days, pain is what they know. It sculpts them, cauterizes them, grows them - but some it destroys. They grew up watching others suffer. In the South Pole, two children were powerless as their mother died. They found out, tears streaming down their faces, but shouldered the burden. Katara carried the guilt for years, holding it inside as a coal to keep her moving forward, reminding her that it was all her fault. She could have stopped it, it was all her fault. Sokka should have been there to protect his mother, he should have never left her side. Katara took that pain and became the mother figure, Sokka took that pain and became a warrior. They were both still children. They never should have had to watch their father sail away. They never should have had to hold back sobs of hunger when there wasn't enough food and they were supposed to provide for the village. They never should have had to watch babies die because of the cold or because of starvation. They took that pain and grew.
In the middle of the Earth Kingdom, a blind child was born, delicate gentle and sweet. She was treated as fragile, so fragile that even a wrong gust of wind could make her fall apart, and she was hidden from view. Who would ever want to admit that they had spawned a blind weak earthbender? It was heartbreaking to listen as her parents tried to make their friends forget about her. It was infuriating to be treated as weak, and so she became strong. She took the ache in her chest and the tears in her eyes and transformed them into earthbending far beyond her years. She took that pain she'd grown so accustomed to and threw it back.
The Firelord was even more ashamed of his offspring than Toph's parents but he could never hide his son. His son would live forever in the public eye, regardless of the fact that Ozai wished he would cease to exist, and he made no effort to hide that from his impressionable boy. Zuko was scoffed at, pushed into embarrassment, manipulated, and publicly humiliated so often he wondered what he had done to deserve it but realized that there must be something wrong with him and it really was all his fault. He was abused and convinced that he had to be better, always always better. His own father burned and disowned him, and so he had to be better. His father would be glad if he died, and that wrenched his heart out. It hurt to be so hated. He was too young to hurt so badly. He took that pain though and transformed it into a scythe, cutting his enemies down.
Azula, the namesake of Firelord Azulon, the jewel of Ozai's eye, grew up in pain. Her mother feared so badly for her that her love came out as stunted and forced. Ozai put so much pressure on her that she herself came out stunted and fed on manipulation. What else is there to a child who has never known a world without harsh punishment and being pushed to always be the best. She had to be in control, but she had to listen quietly to Ozai. He owned her and hurt her, turning her into a weapon making her believe she was unworthy of actual loyalty, the kind that comes without threats. That pain destroyed her. Being so completely alone destroyed her.
They were the generation that grew up on pain. It's all they can ever know.