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.

Mission	Rhapsody
Flames of Prayer	Rhapsody in White
A Land After Time	Rhapsody in Umber
The Lost Avatar	Rhapsody in Azure
From the Ruins	Rhapsody in Crimson
The Man Behind the Mask	Rhapsody in Mauve
The River Runs Red	Rhapsody in Fuchsia
Penance	Rhapsody in Puce
Nary a Cloud in Sight	Rhapsody in Ochre
The Orb's Radiance	Scintillating Rhapsody

A list of items each unlocks can be found under Curio Vendor Moogle.

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.