Organic usually means that no pesticides, growth hormones, extra chemicals, preservatives, injections of anything, gassing, etc are used to produce and/or process the food. It's healthier in which you're not eating a bunch of man-made chemicals and many of the food items still retain their full vitamin and mineral counts.
I prefer it. It actually has real food taste. Everything else is either bland or tastes like salt. I'm the only one who eats it, although my dad will join me when it's produce (he describes supermarket tomatoes as 'baseballs' in texture and taste).
The problem is that the term 'organic' is used by non-green companies to sell their same products with a different packaging for a higher price - tapping into the market demand, of course, and not really caring that they're green or not.