Variants are alternate versions of stories (tales, legends, or ballads) which have evolved from the original telling. Variants are often reflective of the location or time in which they are performed.
Examples:
Folk tales like Cinderella or Blue Beard.
Ballads such as The Twa Sisters or Lord Randall.
How are they organized? Variants are often organized by tale type. Tale types are categorized based on common motifs or themes. They are sometimes organized under the term migratory legends (i.e. Christiansen Migratory Legend Type or Aarne–Thompson–Uther Index).
Examples:
Supernatural legend tale types like (ML 6000) Tricking the Fairy Suitor, (W40A / MLSIT 3036) Witches Sink a Ship, or (ATU 505) The Grateful Dead.

Image: A scan of a manuscript for the 1807 printed edition of Jack and the Beanstalk. Printed for E. Tabart at the Juvenile and School Library, London. Image copied from Wikimedia commons.