The amniotic sac, together with the amniotic fluid it contains, is the same thing as your waters or bag of waters (you know, the one that breaks when you go into labor). The sac consists of two membranes (known as the amnion and the chorion) that form baby's home while in your belly. This keeps baby warm, safe and surrounded by cushiony liquids.
These membranes usually rupture when it's time for baby to make an arrival, and may trickle or gush out of your body, signaling that it's high time to give your doc or midwife a call. In certain cases, if your amniotic sac has not broken and your doctor sees the need to induce labor, she may choose to rupture them with a special tool in order to start contractions and get the show on the road. Most women will go into labor within a few hours after this is done. However, if your amniotic sac gives way well before labor begins, other methods of induction might be used. (Baby can't hang around for too long sans amniotic fluid without risk of infection.)
Expert source: American College of Obstetrics and Gynecologists. Your pregnancy and birth. 4th ed. Washington, DC: ACOG; 2005.