Not coming from a Catholic background, I was always a little naive about catholicism. I thought the Papacy was a hereditary position, like being a king. It only made sense to me that John Paul II was John Paul Senior's son. Turns out, of course, that the Popes don't tend to sire offspring, and new popes are actually elected, as discussed previously.
There is some fascinating history surrounding the office of the Vicar of Christ- it seems that a few popes actually did reproduce. Pope Innocent VIII was not that innocent and fathered several illegitimate children. Pope Julius II had a couple of somewhat notorious children- Lucrezia and Cesare Borgia, whose various lovers, spouses and rivals had a tendency to die unnatural, untimely and violent deaths.
Perhaps most intriguing, if not 100% historically certain, is the story of Pope John VIII, who ruled from 853-855 A.D. John's abbreviated reign was due to the birth of a child. What makes this one so notable is that it was John who gave birth. "John" was actually a woman, and is sometimes called Pope Joan.