Katara first met Toph in Gaoling when Aang was trying to convince her to become his earthbending teacher. It was difficult at first but they eventually succeeded. When the team sat up camp for the first time, Katara was annoyed that Toph was not helping because she sat up her own belongings. She was also disgusted of Toph's tomboyish behavior. The two eventually argued when they were being chased by Azula's team because Katara blamed Toph for not helping her unpack which eventually led to her leaving the group. They eventually worked things out and started to get along better.
While they were in the desert, Appa was captured and Toph could not protect him because she was preventing the library from sinking when the rest of Team Avatar was inside. Aang was enraged and blamed Toph for letting the sandbenders capture Appa so Katara came to her defense and assured Aang that Toph did all she could and that it was not her fault. Later, the group fought off some Buzzard Wasps and Katara helped Toph aim her earthbending since she could not see the enemies in the air.
When Team Avatar was in Ba Sing Se, they decided to seek out the Earth King at a birthday party for his pet bear. Toph decided that Sokka and Aang could not learn proper manners but taught Katara how to behave properly in high society. She was successful and they both managed to enter the party. Days later, Katara decided to take Toph out for a girl's day at the Fancy Lady Day Spa. Toph was reluctant but they ended up having a good time. A group of girls later came by and made fun of Toph. Katara tried to get Toph to ignore them but they ended up knocking them in the canal instead. Katara told Toph that she is beautiful and Toph wanted to return the compliment but she could not see.
Months later, Team Avatar was traveling through the Fire Nation in disguise and a conflict started to arise between Katara and Toph. They were helping Aang practice his water and earthbending but they ended up fighting each other instead. Toph started to scam people in town and Katara felt unsafe about this. She was especially enraged when she discovered a wanted poster of Toph. However, Katara changed her mind when Toph described her as "un-fun" and heard her talking to Sokka about how motherly she is. Katara came up with a scam to turn in Toph for the bounty and have her escape with metalbending. However, they were both imprisoned in a wooden cell instead. Toph revealed that she was upset for hurting her parents and Katara comforted her. After they escaped, Toph had Katara help her write a letter to her parents.
Katara and Toph continued to be very good friends following the Hundred Year War. Katara was happy to see Toph when she arrived in Yu Dao to help them. Sokka always complained about "oogies" when he saw Katara with Aang but Toph usually defended them and remarked that Sokka also demonstrated "oogies" when he was with Suki. At one point, Toph licked Katara's hand in order to demonstrate "oogies" which slightly bothered her.
Katoph has a mild amount of popularity among the Avatar fandom. Many ship them together because they see them as having opposite personalities. A lot of fans cite the fact that Katara called Toph beautiful and the fact that Toph eventually admitted that she was fun. Many also love the moments when Katara tried to comfort Toph when she was being bullied and when she finally admitted that she was upset over hurting her parents.