It is neither. The Catholic Church is not a democracy nor a republic since it was not founded by human beings like secular nations are.
It was founded by Jesus Christ, who personally entrusted the fullness of His authority to Saint Peter and his successors. Christ also commissioned the apostles to help govern the local churches. Their successors, the bishops, work in union with and in obedience to the successor of Saint Peter, the pope (bishop of Rome).
Lay parishioners do not elect their leaders (pastors) as they would in a democracy, and priests do not choose their bishops. The authority of the Church comes from God, who entrusted it to the Hierarchy of the Church (the pope and bishops).
The Church is not a monarchy, either, because the pope does not name his successor. The College of Cardinals elects the new pope. The Catholic Church is unique in that other religions are governed by synods (councils of bishops), boards, or committees.
Catholicism, however, depends on the Petrine ministry (the papacy) in conjunction with the local bishops and the parish priests to effectively serve the spiritual needs of God’s people.