Rhapsodies are obtained by progressing in Rhapsodies of Vana'diel Missions. They are permanent key items whose effects are always active. A complete list can be found here: Square Enix.

They have a dramatic impact on your life in Vana'diel, granting a wide variety of bonuses, reduced wait timers for events, and discounts, as well as increases to the number of Alter egos that can be summoned at one time.