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 loss 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.
Tigress: He tried to take the Dragon Scroll by force, and Shifu had to destroy what he had created...but how could he?
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.
The expression on Shifu's face when Tai Lung breaks Oogway's staff.
Po's real back-story. He grew up in a village filled with pandas when suddenly Shen and his army invaded and started killing any panda that stood in their way. His father presumably died protecting his son and wife whom he told to run away. His mother, chased by a couple of wolves, manage to stow her baby in a radish container, give him one final smile, and left before presumably getting attacked by Shen's men. Up until that point, he had no idea why the pattern on Shen's feathers caused him to have a Heroic BSOD.
Any of the bullying the Furious Five do at Po, particularly how cruelly they talk about him behind his back, and how Po just puts on a brave faceand endures it. Especially how Po snaps at Shifu towards the end when he tries to make Po fight Tai Lung when he first strikes and Po fires off a small tirade about how no one believed in him, the FF and him especially.