Broadly speaking, plainsong is the name given to the body of traditional songs used in the liturgies of the Catholic Church. The liturgies of the Orthodox Church, though in many ways similar, are generally not classified as plainsong, though the musical form is nearly as old as Christendom itself.
Plainchant, or chant, is the music of the medieval Christian church.
As plainchant developed in the West, local traditions emerged in Spain (mozarabic), Ireland (Celtic), France (Gallican), and several in Italy (Milan, Benevento, Ravenna, Rome).
During the Carolingian renaissance (750-850), one specific form of chant probably elaborated in Rome was introduced throughout Western Europe; it developed and progressively displaced other chants: this is Gregorian chant, which remains the official chant of the Catholic church.