Cathedral
A cathedral is the home church for the bishop or archbishop of a diocese. It takes its name from the bishop’s chair, called a cathedra in Latin. The cathedra represents the position and authority of the bishop, and the place where he resides in the territory of his jurisdiction. The bishop is the pastor of his cathedral parish and appoints a rector to manage its spiritual and temporal affairs. It is the main church of a diocese and functions as a parish church within itself. The Cathedral of the Arlington Diocese is Saint Thomas More.
Basilica
A basilica is an important church building designated by the Vatican because it carries significant historical, spiritual, and architectural qualities and attributes. The designation of Minor Basilica is the highest permanent designation for a church building, and once a church is named a basilica, it cannot lose its basilica status. The three signs that a church has been named a basilica are the Ombrellino (umbrella), Tintinnabulum (Bells), and the Papal Cross Keys.
Shrine
A shrine is a church or other sacred place where either a relic is preserved or where an apparition or historical event of faith has taken place. A shrine is typically designated to help foster devotion to a particular saint. For example, the Basilica Shrine of the Immaculate Conception (a basilica and a shrine) was built to foster devotion to our Blessed Mother in the United States, particularly since she is the patroness of America under the title of the Immaculate Conception. Shrines are regulated by the local bishop, and national shrines are designated as such by the United States Conference of Catholic Bishops (USCCB).
Can a church be a basilica, a cathedral, and a shrine?
Yes, a church can be a basilica, a cathedral, and a shrine or any combination of those three. For instance, the Basilica of the National Shrine of the Assumption of the Blessed Virgin Mary in Baltimore, Maryland is not only a basilica and a shrine, but also the co-cathedral of the Archdiocese of Baltimore and the first Catholic cathedral in the United States.