Not much is known about Delly, but what is known is that she is very nice and has been a friend of Peeta ever since they were little children. She used to tell people that he was her brother. Delly was also in the same class as Katniss and Peeta, and was said to admire Katniss. Katniss mentions that Delly wasn't really fat, but she did have some spare weight as a child.
The Hunger Games
When Katniss and Peeta were having a meal in the Capitol along with Haymitch, Cinna, Portia, and Effie Trinket, the Avox Lavinia waited on them. Katniss claimed to recognize Lavinia but after Effie and Haymitch explained that Avoxes are traitors, Peeta chimed in claiming she just looked like someone they knew from District 12, Delly Cartwright. Lavinia and Delly Cartwright look nothing alike in reality but Katniss was grateful for Peeta's discretion.
Delly and her little brother survived the bombing of District 12. Her parents died in the bombing. Later she and the remaining population of District 12 made it to District 13. She was the first person to see Peeta after he was rescued and was one of the only people who could calm him down, chosen to talk to him as she was one of the few people Peeta knew who had no direct connection to Katniss, thus ensuring that the sight of her wouldn't provoke another hijack-induced outburst. Delly visits Peeta and tries to get his memory back. She tried to convince him of all the good things Katniss did. Peeta is happy to see her and asks her where he is and why his family has not come to see him. Delly becomes sad and tears up when she tells him that District 12 was burned down and that his family is dead. Peeta is enraged and shouts that it is Katniss's fault and that she is a muttation.
Delly meets Katniss, Gale, Beetee, Johanna, Finnick, Annie and later Peeta at lunch. When Peeta was allowed to eat with everyone, he was rude to Katniss and said he might try to steal Annie away from Finnick, not caring that Katniss was there. After Katniss and Gale left, Delly was very angry at Peeta and yelled at him for treating Katniss so badly.
Delly is not mentioned after the attacks on the capitol were over. She most likely stayed in refuge in District 13, until District 12 was cleared out of old ruins.
Delly has a younger brother who also survived the bombing of District 12. Her parents owned a shoe shop in town near Peeta's families bakery, which was burned down by the bombing in District 12. Her parents were killed in the bombing of District 12.
In The Hunger Games, Delly Cartwright is described as a pasty-faced, lumpy girl with yellowish hair.
In Mockingjay she is said to look older with her long blonde hair in a practical braid instead of curls. She's also a bit thinner than when she was in District 12, because of the grief from her parents' death. Katniss said she remembers Delly being one of the few kids in District 12 with a few pounds to spare. She and her younger brother had survived the bombing of District 12.