My husband and I were both previously married. He is Presbyterian and I am Catholic. Does he have to become Catholic in order to be married in the Church? Will we have to have previous marriages annulled before marriage in the Catholic Church?

It will be necessary for both of your previous marriages to be declared null before you can have your current marriage blessed (convalidated) in the Church. Permission is required for a Catholic to marry a non-Catholic, but there is no requirement for the non-Catholic party to become Catholic.

The Catechism speaks to couples in a situation such as yours and encourages you to seek forgiveness and live chastely:

If the divorced are remarried civilly, they find themselves in a situation that objectively contravenes God’s law. Consequently, they cannot receive eucharistic Communion as long as this situation persists. . . . Reconciliation through the sacrament of penance can be granted only to those who have repented for having violated the sign of the covenant and of fidelity to Christ and who are committed to living in complete continence. (CCC 1650)