"[Mowgli] came naked, by night, alone and very hungry; yet he was not afraid!" (1.44)
We're not sure if Mowgli's appearance is necessarily bravery or not. Perhaps Mowgli is just too young to know he's in danger.
"I have the Pack and I have thee; and Baloo, though he is lazy, might strike a blow or two for my sake. Why should I be afraid?" (1.74)
However, as Mowgli gets older, we see he's led a life of relative safety. This could be deadly, making Mowgli falsely confident, believing he can face dangerous situations on his own, when he really can't.
"Why should I fear? I remember now—if it is not a dream—how, before I was a Wolf, I lay beside the Red Flower, and it was warm and pleasant." (1.109)
Mowgli seems to have a dim memory of fire, which makes him more confident handling it than any of the animals are. They are all scared of fire.
"No one, then, is to be feared," Baloo wound up, patting his big furry stomach with pride. (3.18)
Baloo teaches Mowgli to communicate with other animals, and this skill opens up the jungle to the boy, giving him the power to communicate instead of fear the animals that are his neighbors.
Bagheera heard, and the cry that told him Mowgli was safe gave him new courage. (3.119)
The only reason Bagheera is fighting the monkeys is to recover Mowgli. If Mowgli were a lost cause, though, Bagheera might not find the fight worth it.
"A brave heart and a courteous tongue," said [Kaa]. "They shall carry thee far through the Jungle, Manling." (3.141)
You could replace "Jungle" with "life" here, and have a pretty apt analogy for how far Mowgli's bravery will carry him.
[Sea Catch] was scarred all over with the marks of savage fights, but he was always ready for just one fight more. (7.3)
Kotick's father is a brave seal, and never one to back down. Clearly his son inherits this characteristic, and then some.
"I am going to follow," [Kotick] said, and his eyes nearly popped out of his head as he shuffled along in the wake of the herd. (7.37)
If a bunch of your friends were blindly following some people to their death, what would you do? You might run the other direction and look for help, but Kotick just follows them to see what's going on. That's either brave or stupid, we're not sure which.
Elephants who were afraid always got hurt […] so before [Kala Nag] was twenty-five he gave up being afraid, and so he was the best-loved and the best-looked-after elephant in the service of the Government of India. (11.1)
We normally associate courage and bravery with doing something for someone else—saving a cat from a burning building, saving someone from drowning—but in this case, we see that Kala Nag the elephant let go of his fear for himself and himself alone. He wanted to live a better life, and that's best accomplished for an elephant by being unshakeable.
"Generally I have to go in among a lot of yelling, hairy men with knives." (13.38)
The animals of "Her Majesty's Servants" are only brave because they have to be. They've basically been ordered to do so, much like the soldiers they serve.