There are many different types of stories. The most important consideration when choosing a tale to tell is whether you like it enough, to tell it with enthusiasm. Stories should communicate to you, a need to be told.
Some of the different categories of stories available to storytellers are: —

Fable - a short moral story not based on fact, using animals as characters, such as, Aesop's Fables - The Fox and the Grapes, Lion and the Mouse and others.
• Fairytale - The best-known would be Grimm's fairytales, about imaginary folk, such as elves, giants, witches, gnomes, and fairies. Closer to home is Mary and the Leprechaun, by Irish-Australian writer, the late John Kelly.

• Folk tale - a traditional story, in which ordinary people gain special insight, transforming them and enabling them to overcome extraordinary obstacles. See The Magic Orange Tree & other Haitian Folktales, by Diane Wolkstein.

• Legend - a story based on the life of a real person, in which events are depicted as larger than life, for example, The Stories of Robin Hood, or King Arthur and the Knights of the Round Table.

• Myth - a story about gods and heroes, explaining the workings of nature and human nature. See Psyche and Eros, or Inanna, by Diane Wolkstein.

• Parable - a fictitious story told to point to a moral, for example, The Sower and the Seed, from the New Testament, of The Bible.

• Personal story - a life story from your own or your family's experience; such as, Streets and Alleys, by Syd Lieberman.

• Religious story - an historical and philosophical story, based on a particular culture and religious persuasion; for example, The Story of Lazarus, from the Bible.

• Tall tale - an exaggerated story, often humorous. Fishing stories, Australian Bush stories, see The Loaded Dog, by Henry Lawson.

• Traditional tale - a story handed down orally, from generation to generation, such as the Polynesian stories - Maui, and The Coming of the Maori.

From About Storytelling - Published by Hale and Iremonger, Sydney, Australia — ISBN 0 86806 593 5 - by Helen McKay and Berice Dudley © 1996