This symbolism is sort of explained in the quotation from Genesis found before the preface in Twilight:
But of the tree of knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. Genesis 2:17
I'm not going to go into the whole story of Adam and Eve. Suffice it to say, God created a garden, created animals and created two humans (named Adam and Eve). The humans were allowed to eat any fruit or vegetable they wanted, except the fruit from the "Tree of Knowledge of Good and Evil." The humans ate the fruit from that tree, disobeying God. God threw them out of the Garden. The end. If you want the more detailed version of the story, consult a Bible.
The Forbidden Fruit is traditionally depicted/thought of as an apple, at least in the Western world. Presumably, Meyer means for the apple on the cover of Twilight to symbolize Bella and Edward's "forbidden love." Of course, what is so "forbidden" about Bella and Edward is anyone's guess. The Cullens, with the exception of Rosalie, immediately accept Bella, as does Charlie with Edward. Nobody at school raises any serious objections.... The only ones who keep complaining about how "dangerous" and "forbidden" their love is, are Bella and Edward. As the series progresses, the only objections raised to Bella/Edward come from Jacob (who is really just jealous of Edward), and the Volturi, who simply don't want a human knowing about the vampire world.