Afterbirth is never really talked about. In my experience, either way, the doctor always made sure everything came out that was supposed to. It was uncomfortable when she would reach in with her hands. If the placenta is still attached you could bleed out. That's why they have to make sure it comes out.
here's a nifty little animation of childbirth where they show you the placenta being delivered at the end (it's a computer animation so it's not graphic or gross, I promise, lol). http://www.youtube.com/watch?v=B84MewU8h7Y
agree with Alaysha - the cord is actually the link between the baby's system and the placenta, which is a specially-grown organ that attaches to your uterus and nourishes the baby while it's growing in your womb. After the baby is born, they will cut the cord and shortly after, you will deliver the placenta (with the rest of the cord attached). If you have ever heard of the "after-birth"...that would be the placenta.
Its connected to the placenta which comes out of you shortly after baby