Once there lived a bamboo cutter and his wife who longed for children. One night the woman asked the great Mountain Fujiyama to send a little child to comfort her. And from the top of the mountain, there was a gleam of light. The woman called her husband, who went to the top of the mountain and found a "tiny moon-child, fragile, dainty, radiant, clad in flimsy, filmy moonshine, more beautiful than anything he had ever seen before." The child told the bamboo cutter than she was Princess Moonbeam, daughter of the Moon Lady, sent to Earth to comfort the couple. The man took her home and the Moon-child gave them comfort and joy for many years.
Note: this summary and quotations were taken from "My Bookhouse Through Fairy Halls", first published in 1920. The details of the story vary from other versions, like this online version.
Fun fact (from wikipedia): In The Neverending Story by Michael Ende, "Moonchild" is the new name that Bastian gives to the Childlike Empress, thus saving the world of Fantastica. It drove me crazy that you could never understand what he says at the end!