A caul (Latin: Caput galeatum, literally, "head helmet") is a thin,
filmy membrane, the amnion, that can cover a newborn's head and face
immediately after birth.
To be "born with the caul" means a child is born with a portion of the amniotic sac or membrane remaining on the head. There are two types of cauls. The most common caul is adhered to the head and face, and looped around the ears of the infant. The lesser occurring caul drapes over the head and partly down the torso of the child. In Germany, this would be called a "helmet" [Galea] for boys, and in Italy, for girls, a "fillet" [vitta] or "shirt" [indusium, camisia].
The caul is harmless and is easily removed by the doctor or midwife. If done correctly, the attending practitioner will place a small incision in the membrane across the nostrils so that the child can breathe. The loops are then carefully un-looped from behind the ears. Then, the remainder of the caul can either be:
If removed too quickly, the caul can leave wounds on the infant's flesh, which may leave permanent scars.
The caul membrane in most cases will be preserved and given to the mother. However, the parents may or may not be told that their child was born with the caul. This depends upon the particular practice of the hospital.
Those born with the caul are referred to as "caulbearers", and are normally noted as possessing the gifts of clairvoyance and other types of "supernatural" abilities. 
The "en-caul" Birth
The "en-caul" birth, not to be confused with the "caul" birth, occurs when the infant is born inside of the entire amniotic sac. The sac balloons out at birth, with the child remaining inside of the unbroken, or partially broken membrane.
Being born with a caul is rare, occurring in fewer than 1 in 80,000 births. This statistic includes "en-caul" births, which occur more frequently than authentic caul births; therefore authentic caul births are rarer than the statistic indicates. Most "en-caul" births occur in premature births.
In medieval times the appearance of a caul on a newborn baby was seen as a sign of good luck. It was considered an omen that the child was destined for greatness. Gathering the caul onto paper was considered an important tradition of childbirth: the midwife would rub a sheet of paper across the baby's head and face, pressing the material of the caul onto the paper. The caul would then be presented to the mother, to be kept as an heirloom. Some Early Modern European traditions linked being born with the caul to the ability to defend fertility and the harvest against the forces of evil, particularly witches and sorcerers. Over the course of European history, a legend developed suggesting that possession of a baby's caul would give its bearer good luck and protect that person from death by drowning. Cauls were therefore highly prized by sailors. Medieval women often sold these cauls to sailors for large sums of money; a caul was regarded as a valuable talisman.
I was born with a caul, which was advertised for sale, in the newspapers, at the low price of fifteen guineas. Whether sea-going people were short of money about that time, or were short of faith and preferred cork jackets, I don't know; all I know is, that there was but one solitary bidding, and that was from an attorney connected with the bill-broking business, who offered two pounds in cash, and the balance in sherry, but declined to be guaranteed from drowning on any higher bargain. Consequently the advertisement was withdrawn at a dead loss ... and ten years afterwards, the caul was put up in a raffle down in our part of the country, to fifty members at half-a-crown a head, the winner to spend five shillings. I was present myself, and I remember to have felt quite uncomfortable and confused, at a part of myself being disposed of in that way. The caul was won, I recollect, by an old lady with a hand-basket.... It is a fact which will be long remembered as remarkable down there, that she was never drowned, but died triumphantly in bed, at ninety-two. (Charles Dickens, David Copperfield, published London 1850)
In Betty Smith's classic novel A Tree Grows in Brooklyn Francie Nolan is born with a caul. The midwife who officiated the birth stole the caul and later sold it for $2 to a sailor from the Brooklyn Navy Yard. It was believed that whoever wore a caul could not drown.
In the film Oscar and Lucinda, Oscar's father gives him the caul that was upon his head at birth. Oscar has a phobia of the ocean and of water in general, linked to the death of his mother when he was a child. He carries this caul with him until he dies, ironically, by drowning.
In the play Gypsy: A Musical Fable, Mama Rose tells Louise (Gypsy Rose Lee): "You were born with a caul. That means you got powers to read palms and tell fortunes - and wonderful things are gonna happen to you."
Another myth associated with a caul is featured in the short story, "The Scarlet Ibis." When the main character's brother, Doodle, is born in a caul his aunt states that cauls are made of Jesus' nightgown and everyone must respect Doodle as he may become a saint someday.