It is important to distinguish between the Byzantine Catholic Church and the Byzantine Rite. The Universal Catholic Church, of which we are a member, is a communion of 23 churches, including the Roman Catholic Church, the Byzantine Catholic Church, and others. The churches are in full communion with each other and are distinct from Orthodox Churches, which are not in full communion with the Universal Catholic Church.
A 'Rite' is a liturgical tradition that describes the celebration of the Divine Liturgy, the Eucharist and other Mysteries (sacraments), and the general spiritual life of the Church. Within the Catholic Church, six distinct rites are celebrated; among them are the Latin Rite and the Byzantine Rite.
Note that many Orthodox Churches celebrate the Byzantine Rite of spirituality. All Byzantine Catholics follow the Byzantine Rite, but not all followers of the Byzantine Rite are Byzantine Catholic; nor are all followers of the Byzantine Rite in full communion with the Universal Catholic Church.