From the Super Mario Wiki, the Mario encyclopedia
The Owl is a character from Donkey Kong 64 that lives in the big tree in Fungi Forest. He wears a green racing shirt and challenges Diddy to a race if he using the Rocketbarrel Boost move. Like the Rabbit, the Owl must first be awoken by a musical instrument move, in his case the Guitar Gazump, before he can be raced.
As would be expected from an owl, he can only be encountered at night. The Owl does not want to talk to anyone who cannot fly, which he says to Diddy if he is not wearing his Jetbarrel. After putting them on, the Owl, in similar vein to the Necky in Angry Aztec, challenged Diddy to fly through that the rings he left behind. Along the way, Diddy must avoid hitting into the tree branches, and if Diddy misses three rings, Diddy loses at the challenge. When Diddy completes the course, the Owl feels disappointed that his obstacle course was not as hard as thought it was. The Owl then rewarded Diddy with a Bonus Barrel at the top of the tree (unlike most characters, which reward a Golden Banana for completing an objective directly).