Yoshi's Island, also known as Super Mario World 2, is a game released in '95 for the SNES. It's a Mario platformer where you play as Yoshi! It has over 3 dozen levels, and features such as transforming, and having Baby Mario on your back. This will be explained shortly.
A stork flies overhead, carrying Baby Mario & Baby Luigi. It is attacked! Baby Luigi is stolen, but the stork drops Baby Mario. Baby Luigi is taken away while his brother falls...
Mario lands near a Yoshi. That Yoshi calls it's friends and they puzzle over this for a time. Then, they decide to return it to it's family by carrying him on the Yoshi's backs, fight all of Bowser's minions he plans to stop you with, and rescue his brother. Welcome to Yoshi's Island...
This game is that of a platformer, but very different from Super Mario Bros. You have these new abilities that will be very useful to get to the end: Eating enemies and making them into eggs, the Flutter Jump, transforming, and many more. You also have Baby Mario to deal with. Each time you're hit by an enemy, Mario will start floating around in a bubble and crying. Until you get him back, you'll lose one Star Point per second. When your SP reaches 0, some flying enemies will take the Baby away and you'll lose a life. If you can get him back, however, you'll regain SP gradually if it went under 10 until it reaches 10 again.
Your main attack will be eating enemies and turning them into eggs, which you can throw at enemies to kill them. There are a few different types of eggs that you can get, all of which are different.