A classic.
In Yoshi's Island you play as many different yoshi's, who's main purpose is to make sure that baby mario doesn't get captured by the koopa people, and makes it back home to his brother luigi.
When you play as the yoshi, instead of loosing the powerup you have or dying when you hit a spike, or something attacks you, you actually lose baby mario and a timer which at default is 10 (but you can make it higher by collecting stars or going through the checkpoints) when the timer reaches 0 the koopa type people (actually I'm not really sure what they are) capture mario, and take him to the wizard guy that is in this game and super mario world 1 and 2.
But you might say "Where is Bowser?" Bowser is in this game, but like mario he is a baby. But don't worry, he um, grows. And it is also one of the coolest boss fights I have ever played on a SNES.
Just go get this game, you can probably get it at a garage sale for 1$ (or euro or whatever), or at a pawn or secondhand game shop for 2-5$