Volume 2, Book 23, Number 441:
Narrated Abu Huraira :
Allah's Apostle said, "Every child is born with a true faith of Islam (i.e. to worship none but Allah Alone) but his parents convert him to Judaism, Christianity or Magianism, as an animal delivers a perfect baby animal. Do you find it mutilated?"
Sometimes the first part is rendered "Every child is born in a natural state of righteousness (fitra)" and it is probably one of the most common "proof-texts" for the fact that Muslims don't believe in original sin.
Contrary to a widespread impression, Original Sin is actually not the claim that babies born today get punished for the sins of Adam and Eve. The central core of the doctrine is actually the idea that as a consequence of what Adam and Eve did, people born today are inclined towards sin. That human nature is so fundamentally twisted and perverted, that we can't help but live sinful lives. That's a part of why many Christians say you need to be "born-again" (apparently you didn't get it right the first time). In contrast, Islam teaches that we are born whole, unmutilated, naturally inclined to submit to God's will. Or at the very worst, neutral.
In past conversations I've had with Christians on this subject, this is the point where they typically ask: If there is no original sin, then why is there evil in the world? Good question. One of the oddest statements on this topic ever made has got be the words of Anne Frank: "...in spite of everything I still believe that people are really good at heart."
I would give the example of a family where everyone on some level loves one another or at the very least is well intentioned towards each other, but nevertheless baggage accumulates, and misunderstandings develop. Family members acting out historical events in the past behave in ways which continue to hurt their own loved ones. No one wants to hurt the other, but in spite of that fact, they do.