When Po tries to leave only to be stopped by Shifu who realizes he has to train him. Po confesses that he endured all the abuse Shifu and the Five shoved at him because he is ashamed of being a fat, useless failure and could not abandon his one chance to become something more. It gives the film such heart while Shifu struggles to find a way to help the panda.
Shifu: Then why didn't you quit? You knew I was trying to get rid of you, yet you stayed!
Po: Yeah, I stayed. I stayed because everytime you threw a brick at my head or said I smelled; it hurt, but it could never hurt more than everyday of my life just being me. I stayed because I thought if anyone can change me, can make me not me, it was you! The greatest kung fu teacher in all of China!
When Tai Lung confronts Shifu. The snow leopard's first line, "I have come home, Master," is oddly sincere, even regretful. This of course is followed by Shifu's visibly painful reply: "This is no longer your home... and I am no longer your master." Even during the fight itself there were moments which tugged the heartstrings. The raw pain and anguish in Tai Lung as he cried out how he only wanted to make Shifu proud of him could only be equaled by Shifu's brokenly whispered reply that he was always proud of him. That moment, that one tiny moment, when after Shifu apologized for having failed him, Tai Lung almost, almost relented and did a Heel Face Turn. The relationship between these two has to be one of the most deep, complex in the movie.
Shifu's panic at the lose of his mentor right before Tai Lung arrives is his Darkest Hour. The worst mistake of his life is coming back to beat him into the ground, he has no idea how to make it right and his mentor is no longer around to guide him.
The moment, in flashback, when Shifu could not attack Tai Lung because he in turn flashbacked to him as a cub, was also poignant and emotional.
Oogway's talk to Po (and later Shifu) underneath the Peach Tree of Wisdom.
Similar to Tai Lung, Tigress' backstory. Especially since her past actions made her the jerk she was in the movie.
Tigress' backstory is made worse by the reveal in Secrets of the Furious Five that she was raised in an orphanage where she was the only predator and Shifu was the first person who wasn't afraid of her.
Shifu: I am -
Young Tigress: Afraid?
Shifu:(closes the door) No.
Young Tigress: Well, you should be! I am Tigress! Tigress the monster! ...A monster no one wants.
Shifu: You are not a monster. You're just a little girl.