I don't remember my exact age, but when I was little my Uncle had a Nintendo. He had the original Final Fantasy and I used to play it for hours when we visited their house. My parents wouldn't let me get one, assuming (correctly) that I would want to play video games and do little else.
I still love the fantasy genre, especially in video games. When I look at the incredible CG, graphics and complex story lines I have to marvel that something so simple as the original Final Fantasy kept me enthralled. I used to love playing Super Mario Bros. 3 with my best friend in junior high as well.
I suppose to me Nintendo means something long-lasting and fun. I have a Nintendo DS today, and I enjoy playing it even though I'm an adult now. I also have an original Nintendo 8-bit system that a friend gave me when he no longer wanted it, and it still works perfectly. I love how something that old can still work after all of these years.
(Oh, and I had the original "brick" Gameboy, a Gameboy Color, Gameboy Advance, Gameboy Advance SP, Gamecube and a Nintendo 64...haven't purchased a Wii yet but if there's an "Animal Crossing" or "Harvest Moon" I may not be able to resist.)