The story for the "Firebird" ballet comes from a different pieces of Russian folklore--as if someone made a Western tale in which a girl wears a red hood, loses her glass slipper and falls into a deep sleep because of an enchanted apple, to be awakened by the kiss of a Prince.
The story follows Prince Ivan as he tries to find the culprit who keeps stealing his father's golden apples. This turns out to be the Firebird, who gives him one of her feathers to use in time of need in reward for releasing her. The Prince then discovers a group of Princesses all kept captive by the evil Katschei, who can only be defeated when someone breaks the egg which holds his heart. With the help of the Firebird to tame Katschei's evil minions, and of course true love, Ivan succeeds in freeing the maidens and securing the most beautiful Princess for himself.
The music for the ballet was composed by Igor Stravinsky. One of his earlier works, there isn't the avant garde aspect of his later music, but this makes it more accessible to the average listener--which is not at all to say the music is boring. Stravinsky used diatonic music (music that uses the regular notes of a traditional major or minor scale) to represent normal, earthly things, and chromatic music (using all the notes available) to represent the world of magical creatures. Listen to Katschei's slithering, evil theme at the beginning. This version seems to have more emphasis on their beautiful set than on dance, but the Firebird comes in and dances around 6:30 (to a wicked hard flute part!). Listen to the violin harmonics (a very high and eerie sounding effect) around 2:00.
You can follow the links to watch the complete ballet. I'm including the last section as well. This starts with the Firebird's lullaby as she enchants the evil minions into a deep sleep and goes into the joyful finale, one of my favorite moments in classical music. Listeners might recognize the music as also being featured in Disney's Fantasia 2000, as the Spring Fairy overcomes the obstacles to spread spring throughout the land (how appropriate for this time of year).