Super Mario World: As a Mario game, I liked this better then SMB3. The ability to fly, Yoshi, the graphics...everything about this game was groundbreaking at the time. Although I am pretty sure everyone who had an SNES played this game (it was a pack-in), it is still essential.
Legend of Zelda: A Link To The Past: This was the game that got me into RPG's seriously, like needing to read strategy guides and such. It took me forever to figure out how to get out of the Ice Palace (you had to pull the tongue of a statue). The storyline was grand and epic, and you really cared about the characters and their world.
Super Metroid: Very under-appreciated for being a beautiful and complex game. The next gen Metroid games should have followed up with this formula, instead of going the FPS route.
Final Fantasy VI: There are a lot of reasons to like this FF, but the two that stick out to me are the villain and the score. Kefka is a evil bastard who forces the world to bend around himself, and the music is the best you will hear on the SNES.
Super Mario Kart: A fun game that you play with your friends, although maybe more influential then it should have been. This first installment was the classic Nintendo formula of creating games which are simple and fun.