The Catholic Church does not forbid Catholics from marrying people who are not Catholic. It has been the practice of the Church to marry non-Catholics and Catholics for quite some time. The Church refers to these types of marriages as mixed-marriages.
Sometimes a future spouse will choose to go through a process called RCIA to become Catholic prior to marriage, but it is not necessary to become Catholic before marrying a Catholic.
However, express permission of the local bishop is necessary. The Catholic person must uphold the obligation to preserve his or her own faith and “ensure the baptism and education of the children in the Catholic Church,” (Catechism of the Catholic Church, 1635).
One of the reasons that the Church exists is to safeguard the souls of those in its care. If a Catholic insists on marriage to a non-Catholic, the Church allows it, but wants to protect the soul of the Catholic in the marriage by making sure the non-Catholic understands the moral teaching and obligations of the Catholic party and assure that the Catholic is not in a position hostile to his or her faith.
Marriage to a Non-Baptized Person
The Catholic Church calls the union of a Catholic to someone who has not been baptized a disparity of cult. In the above example the two people are baptized Christians of different confessions (or denominations), but a non-baptized person is not a part of the Christian family. When it involves someone who has not been baptized then the marriage requires an express dispensation from the bishop in order for the union to be considered valid.
Scripture tells us that the unbelieving spouse is made holy through the believing spouse (1 Cor. 7:14).
Sacramentally in marriage, the spouses are the conduits of grace to each other and in a mixed-marriage of disparity of cult the Catholic is a conduit of grace to the non-believer. If this leads to a free conversion of belief by the non-believer then the church rejoices.
Concerns About Marrying Non-Catholics
For a mixed-marriage between a Catholic and a non-Catholic to work well it is important that the couple embraces what is common between their respective faith traditions and “to learn from each other the way in which each lives in fidelity to Christ,” (Catechism of the Catholic Church, 1634).
This can be very difficult and trying, but can be overcome by mutual respect.