From the Super Mario Wiki
Bonus Challenges are Bonus Games in Super Mario World 2: Yoshi's Island, its remake, and Yoshi's Island DS. When a Yoshi reaches the goal roulette and the roulette lands on a flower, then a Bonus Challenge can be instantly played. In the original Yoshi's Island, a specific Bonus Challenge can also be unlocked and played anytime after getting 800 points in a world; in the later two games, they can be accessed after getting 700 points in a world.
The list of Bonus Challenges are below.
In Yoshi's Island and its remake, the Bonus Challenges available after getting a flower at the goal roulette depends on the world currently being played, as follows: