Yu met Aang for the first time when the Avatar participated in one of his lessons at the academy. Aang was handed a flyer which contained a coupon for one free lesson at Master Yu's Earthbending Academy, and after the actual lesson, where he was beaten in seconds by a younger student, Master Yu suggested that Aang pay for a year of lessons; he further suggested that if Aang paid in advance, he would be bumped up to the next belt. This suggested that Yu was more of a businessman than a true master, as this is a common practice among less-than-reputable martial arts schools. Yu's offer backfired, as it made Aang quickly decide that Yu was not the earthbending teacher he was looking for.
The earthbending instructor met Aang again while he, Team Avatar, and the Beifongs were having dinner at the Beifong Estate. When Aang revealed that he needed to find an earthbending teacher, Lao Beifong was quick to recommend Yu, referring to him as "the finest teacher in the land"; Yu had been hired by Lao Beifong to train Toph in earthbending. However, per Lao's instructions, he only taught her basic moves.
When Aang and Toph were kidnapped by Xin Fu and the other Earth Rumble VI contestants, Lao ordered Yu to go bring back Toph. Yu's contribution to the release of Toph was limited to using his earthbending to slide the ransom money over to Xin Fu. When Toph decided that she would take on the contestants in order to free Aang, Yu was amazed by her skill and commented to Lao that "she [was] the greatest earthbender [he had] ever seen."
Later, he, along with Xin Fu, was hired by Lao Beifong to retrieve Toph, who had run away to join Aang on his travels.
In their search for Toph, Yu and Xin Fu reached the Misty Palms Oasis at the edge of the Si Wong Desert. Master Yu appeared to be the more level-headed of the two as he reminded Xin Fu (who mistook the bounty as dead or alive) that Lao Beifong wanted Toph alive. Xin Fu temporarily changed their target to the fugitives Zuko and Iroh to collect more bounty.
They followed Iroh and Zuko into the cantina where Yu wanted to bide his time, but Xin Fu, ignoring Master Yu's advice, directly confronted the fugitives in the shady bar. This gave the White Lotus member, Fung, with whom Iroh was meeting, the perfect opportunity to create a distraction and turn all the desperate men in the bar against them. Although outnumbered, Master Yu and Xin Fu dealt with them easily with their earthbending skills, but in the confusion the Fire Nation fugitives escaped. They searched the whole town, but to no avail, and finally returned to tracking Toph and the Avatar.
Later on, after luring Toph into a trap with a forged letter from her mother, the duo finally succeeded in capturing her in a metal cage in Ba Sing Se. On their way back to Gaoling, Toph discovered that there were purified earth particles in the metal and she figured out how to bend them, thus discovering metalbending. After her escape from the cage, Toph turned the tables on her captors and trapped them in the very metal box in which they had imprisoned her. Right after being entrapped, Master Yu stated that he needed to go to the bathroom, much to the annoyance of Xin Fu.
Master Yu was somewhat of an easygoing, peaceful man who took pride in training earthbenders at his academy. The idea of collecting a bounty seemed to interest him greatly, as he agreed to aid Xin Fu on a mission to bring Toph back to her parents in exchange for a large sum of money; suggesting that he was a greedy man that taught earthbending for the sake of earning a profit. Master Yu was much more relaxed and patient than Xin Fu and seemed somewhat gullible when he almost fell for one of Toph's tricks that would have allowed for her to escape.
Master Yu was shown to be a proficient earthbending master, and was well known in Gaoling for his abilities. He appeared to prefer a more subtle, conservative approach to his earthbending, as opposed to Xin Fu, who relied on brute strength. Instead of crushing people with large rocks, he would sink them into the ground, which required much less effort.
Yu is most likely the character's surname, not his given name. According to his flyer, his name is written as 余, a traditional Chinese surname.
Master Yu and his academy serve as a satirical parody of typical "strip mall" dojos where parents are able to purchase belts for their children in disregard of the symbolism and substance behind them, as well as a reflection of the disdain Sifu Kisu and Bryan Konietzko have for the use of belt systems.
Master Yu had a very high interest in gold along with his partner, Xin Fu; which prompted them to search for Toph.