“The Nutcracker,” a fantasy-fueled story of a young girl who dreams her toy prince to life and follows him through magical lands, debuted as a ballet in St. Petersburg in the late 19th century with choreography by the ballet master Petipa and music by Tchaikovsky. But it didn’t become an American holiday tradition until after World War II, when George Balanchine staged his version for the New York City Ballet and the ballet caught fire as a Christmastime production.