The Arduino itself isn't really capable of playing music on its own, but add an Anniken Andee shield, and you're good to go!
Arduino models before the Due aren't well known for being able to do audio, but this Arduino-based MP3 Alarm Clock can play lo-fi MP3s.
Due to limitations of the Arduino, music is generally a no-go, However, APC have cunningly managed to get the Arduino to play audio.
