Once you're officially pregnant, your future baby consists of a little ball of swiftly multiplying cells in your uterus. Some of the cells in this bundle will grow into the embryo (which will soon start forming organs), and others will grow into the placenta. So essentially, the placenta comes from the same roots as your child (perhaps explaining why some cultures refer to it as baby's "twin").
And what of this placenta? Well, it's got a pretty big role—it's the vital connection between your body and baby's. Finger-like growths tuck into the uterine walls to connect the placenta to you, while the umbilical cord connects the placenta to baby. Its full functions are complex, but basically it takes goodies like oxygen and nutrients from your blood and moves them into baby's blood to help him stay alive and grow. It also filters waste from baby's blood and dumps it into yours (to be disposed of by your blood). The placenta is the route by which damaging substances like drugs, alcohol or nicotine could reach baby, so remember to stay away from things that could cause harm!
When you deliver, the squishy placenta (also known as the afterbirth) will follow baby out of your belly. If you give birth in a hospital, they will generally dispose of the organ as medical waste, but if you'd like to have a look before it's trashed, just ask—most docs will oblige. Some women choose to keep the placenta and bury it, burn it our even dry it out and take it as a supplement (it's rumored to increase youthfulness and combat postpartum depression).
American College of Obstetrics and Gynecologists. Your pregnancy and birth. 4th ed. Washington, DC: ACOG; 2005.