The Bianchi were one of the many cults of penitents resultant from the various plagues of the High Middle Ages. Unlike the flagellants, the Bianchi sought redemption through peaceful, non-masochistic methods. They are, in fact, quite unique from most other penitent cults of the time period because of their gentle outlook.
However, the story that brought about the inception of the Bianchi was quite typical of the High Middle Ages. In 1399, in the Dauphine province of southern France, a young shepherd supposedly had a vision of the Virgin Mother. She told him that Jesus was angry at the world, and, having already "rained death on one-third the human race" (referring to the Black Death of 1348-1351), he was planning on completely wiping out the human race. The god-fearing men of the world must do penance and then reform others if the Apocalypse was not to come.
The response to this story in Italy was overwhelming. All throughout the summer and fall of 1399 bands of white-clad penitents thronged the streets of urban Italy, singing hymns and urging reconcilliation in all conflicts. The Church, feeling that it had lost control, neither sanctioned nor forbid the marches.
Ironically enough, in early 1400 the plague returned, almost as vicious as it had been fifty years previous.
An interesting study has been done by the Swedish scholar Olaf Brotto about the presence of religion among the masses in the Middle Ages. By looking through parish records he counted the number of religious names used over time and found several trends. The study began in 1260 and continued through the plague years. At first there was great diversity in names, but over time people began to use the names of religious figures more and more. In the generation before the Black Death (early to middle 14th century) religious names suddenly dominated those surveyed; after the Black Death religious names accounted for nearly 80% of those surveyed. Most of the names came from saints associated with the plague, such as Lorenzo (roasted alive on a gridiron as a martyr), Sebastian (a Roman soldier shot through with arrows because of his faith; he is particularly associated with the plague because his arrow wounds look like the notorious buboes that marked infection), and Cosmas and Damian (both patron saints of physicians). It is fairly obvious to see that, following disaster, people turn to religion-- for instance, the rise in religious names in the early 14th century can be accounted for by the Great Famine of that century-- but more interesting to note that cults such as the Bianchi and the flagellants could be so easily frightened into forming. This time period must have been one of religious hysteria, as people sought frantically for answers. It seems almost like the story of Job acted out on a massive scale: the people turn to God, demanding answers, and yet His face remains ever inscrutable, no matter how piteous their protests.
Information on the Bianchi comes from David Herlihy's excellent collection of essays on the Black Death, The Black Death and the Transformation of the West.