Heirloom fruit and vegetables are older varieties that will reproduce exactly the same kind of plant again from its seeds (hybrids can't). Biodiversity is also a very important reason to choose heirlooms over hybrids (and let me not even mention the G word), as we need as many plant varieties around as possible, and especially those that grow well under specific local or strained weather conditions. As a matter of fact, I read somewhere that Peruvians have almost as many potato and corn varieties as growing places, because these plants were all developed for very specific local conditions, and would not perform as well if planted elsewhere. Now that is biodiversity! 
In addition, heirlooms are often more disease resistant and have more intense flavors - think intensely flavorful strawberry or tomato instead of those watery spongy supermarket kinds. There are now even seed libraries to preserve heirloom varieties for future use and generations. Sometimes you can draw from them, but you have to return seeds at the end of the season in exchange for your loan. Lastly, it is infinitely more interesting to taste many different pepper or tomato or apple or carrots kinds (love the purple carrots) than the one or two same old same old you get at the supermarket.
Hybrids, in comparison, while having some desirable characteristics, can't reproduce from their seeds - think of seedless watermelons or grapes.
And by-the-way, heirloom breeds exist among animals as well, and some farmers are now bringing these older breeds back for the same reasons heirloom produce is desirable.
