Adage (French), Adagio (Italian)
In music, Adagio means "slowly", and in ballet it means slow, unfolding movements.
In a classical ballet class, the Adagio portion of the lesson concentrates on slow movements to improve the dancer's ability to control the leg and increase extension (i.e., to bring the leg into high positions with control and ease).
In a Grand Pas (or Classical Pas de deux; Grand Pas d'action; etc.), the Adagio is usually referred to as the Grand adage, and often follows the Entrée. This Adage is typically the outward movement of the Grand Pas where the Ballerina is partnered by the lead male Danseur and/or one or more suitors.
In ballet, the word adagio does not refer to the music accompanying the dance but rather the type of balletic movement being performed. For example, the Grand adage of the famous Black Swan Pas de deux from Swan Lake is musically an Andante, while the choreography is Adagio.