An episiotomy is an incision in the perineum (the skin between the vagina and anus) that helps baby fit through.
Doctors used to routinely perform episiotomies because a surgical cut was thought to prevent tearing. But, it's now known that episiotomies don't prevent tears, and that natural tears actually tend to heal better than episiotomies. However, doctors will still perform an episiotomy in certain situations, such as to help deliver baby more quickly when there are signs of fetal distress.
If your doctor does decide you need an episiotomy, you'll receive a local anesthetic (unless your perineum is already numb from the pressure of baby's head). Once baby is safely delivered, your doctor will give you another shot of local anesthesia and stitch up the cut. After a few weeks, the stitches (and your memory of this icky part of childbirth) will disintegrate.