Myth – A traditional, typically ancient story dealing with supernatural beings, ancestors, or heroes that serves as a fundamental type in the worldview of a people, as by explaining aspects of the natural world or delineating the psychology, customs, or ideals of society. (http://www.thefreedictionary.com/myth)
An example of a myth is Hurcules. Hurcules is the son of Zeus, who is the most powerful god, and one day he gets sent into insanity. He almost killed 3 boys and to cleanse himself, he visits Oracle. Oracle tells him to go to Eurystheus who sends him off to impossible challenges. (http://www.gradesaver.com/mythology/study-guide/section13/)
Legend - An unverified story handed down from earlier times, especially one popularly believed to be historical. (http://www.thefreedictionary.com/legend)
An example of a legend is the legend of the Fountain of Youth. The legend tells about a special fountain which contains water with special powers. It is believed that by drinking the water from the fountain, a person becomes an eternal youth. (http://hassam.hubpages.com/hub/Famous-Legends)
Folktale – a tale or legend originating and traditional among a people or folk, especially one forming part of the oral tradition of the common people. (http://dictionary.reference.com/browse/folk+tale)
A example of a folktale is Bear Lake Monster. The tale tells how there is a Bear Lake Monster at Bear Lake in Utah and about what some people did when they saw the Bear Lake Monster. (http://americanfolklore.net/folklore/2010/07/the_bear_lake_monster.html)
I chose The Tale of the Bamboo Cutter. It is an old Japanese folktale. It is about an old bamboo cutter with out any children and about a Princess Kaguya. One day when the bamboo cutter was walking in the bamboo forest, he found a shining stalk of bamboo. He cut it open and inside was a baby. He raised the baby with his wife and named her Kaguya-hime (Princess Kaguya). After that, every time he cut open a bamboo stalk, out came a nugget of gold and they got rich. A lot of men came to ask if they could marry her but she rejected them because she needed to go back to Tsuki-no-Miyako and the day came and she left.