They are ghosts of deceased beautiful women, who wear white dresses and kidnap people they interact with. When they were alive, they all had children and, when they suffered from a temporal alienation after being rejected or being deceived by the men they loved, they drowned them. Then, they commit suicide and, subsequently, they must spend the rest of eternity on Earth looking for their children, crying and suffering, until they find them.
They roam in rivers and roads, and when they find a man who cheats, they kill them and the men are never seen again. They get angry when someone does not want to obey them, so they attack them while taking their true form.
Even though they can appear as normal and beautiful women in white dresses, they have another form they hide. Their true form bears resemblance to a corpse, grey skin, sunken eyes and no lips. Also their hands become claw-like in this form.
As ghosts, they possess enhanced strength and speed (surpassing that of a human), invisibility and telekinesis, and other abilities--such as being able to teleport or manipulate the area that surrounds them (including flickering lights).
As any other ghost, they are weak to iron (being hit by a weapon or thing made of iron can make them dissapear briefly), and salt (which can be used to stop them temporarily and protection circles). However, none of them are effective to destroy them, but there are two methods to permanently defeat a Woman in White. One of them is salting and burning their bones, a normal method to get rid of ghosts.
The other one, used by Sam and Dean Winchester in one of their cases, was bringing the Woman in White they were fighting to the house where she drowned her children. She was afraid to confront their spirits and never went 'home'. The children's spirits overpower her and drag her back to the afterlife.
The Woman in White could have been taken from La Llorona (sometimes called the Woman in White), a ghost of a woman crying for her dead children from Mexican folklore, that has extended over Latin America's other folklores.
"La Llorona" ("The Weeping Woman") is a popular legend in Mexico, with many versions extant. The basic story tells that La Llorona was a beautiful woman by the name of Maria who killed her children by drowning them in order to be with the man she loved, but was subsequently rejected by him. (He might have been the children's father who had left her for another woman.) Then, after being rejected by her lover, she killed herself. When Maria reached the gates of Heaven, she was asked, "Where are your children?" and she replied, "I don't know, my Lord." She was not permitted to enter heaven until she found her children. She now wanders the Earth for all eternity, searching in vain for her drowned offspring. Her constant weeping is the reason for her name. In some cases, according to the tale, she will kidnap wandering children who look like her lost children or children who disobey their parents.