A folk tale is a fictional story passed down through generations. Every time the tale passes from one generation to the next, it may change slightly. The folk tale tends to be short and often features magical creatures, fantasy kingdoms or personified animals. The folk tale tells a succinct story about people or animals who triumph over adversity and villainy. Peasant class characters in folk tales often make fools of upper class characters. Original folk tales tend to be more appropriate for adults than children, but in modern tellings, many of these same folk tales are modified and are suitable for children.
A fairy tale is a folk tale that features fairies or magic. This subgenre of the folk tale genre is more likely to feature royalty, while the typical folk tale will feature an everyday person or animal. "Cinderella" is an example of a folk tale fairy tale. It features royalty and magic, but it also features an everyday person, Cinderella, who triumphs over the upper class, her stepmother and stepsisters.
All folk tales and fairy tales are stories. The label of "story" applies to many tales, both fiction and non-fiction. A story simply is a tale with a point to it. Any mundane detail can become a story if the storyteller frames the event as a story. Telling someone what you ate for dinner last night is a fact; telling a person the tale of how you decided to order out for dinner because a pipe burst in your kitchen can be a story if you give the listener a step-by-step reenactment of the evening's events.
Not all stories are folk tales simply because the label "story" is so broad. Your recounting of the events of the previous night can be a story, but it's certainly not a folk tale. A folk tale is a fictional story meant to show good win over evil or to teach a moral lesson. The folk tale survives generations. Some stories get passed through some generations, but the vast majority, of course, do not.