The umbilical cord is the connection between the baby and the placenta, throughout the pregnancy. It consists of 3 blood vessels: one carrying oxygen and blood to and from the placenta to the baby, and the other two vessels function by carrying waste from the baby to the placenta.
Shortly after the birth, the umbilical cord is cut, waiting for it to "tap out" is beneficial in letting the transfusion of blood flow from the placenta back to the baby and reduces the risk of anemia (iron deficiency. You can hold the cord between two fingers (be sure to sanitize your hands and wear medical gloves ) and feel it tapping out, it is pretty amazing to experience. Mention this to your gynecologist or midwife before the birth so they can aid you.
Another option is not to cut the umbilical cord and let it fall off on its own, this is called a "lotus birth" (to help the baby transition to life outside the womb without being cut from their mother).
After cutting the umbilical cord, it shrinks and falls off between 7 to 10 days.
An umbilical cord in a cord ring, one day after birth.
During the postpartum period, the umbilical cord falls off and it does not hurt the baby. You can keep the navel clean by using filtered water and patting it dry with a gauze or muslin towel. No worries about bathing your baby during this time, use a neutral or non-aggressive soap. If it becomes irritated be sure to contact your general practitioner.