Who can baptize another person?

It requires no special minister to baptize; however, outside of an emergency or some other necessity, baptism is ordinarily reserved to a bishop, priest or deacon. Thus the Catechism of the Catholic Church provides,

1256. The ordinary ministers of Baptism are the bishop and priest and, in the Latin Church, also the deacon. In case of necessity, anyone, even a non-baptized person, with the required intention, can baptize, by using the Trinitarian baptismal formula. The intention required is to will to do what the Church does when she baptizes. The Church finds the reason for this possibility in the universal saving will of God and the necessity of Baptism for salvation.

Should we baptize in the name of the Father, Son, and Holy Spirit, or in the name of Jesus?

The Catholic Church teaches that baptism is valid only if water is used together with the Trinitarian formula.

Matthew 28:19 “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.”

The use of the name of Jesus has been practiced by various groups down through history, often by those denying the doctrine of the Trinity in some fashion, or the proper character of the “Father,” “Son” and “Holy Spirit” as names for the Divine Persons. Although, St. Peter, and St. Paul use other references for Baptism, such as “in the name of Jesus,” or “into His death,” these are theological statements about what baptism accomplishes. The only clear formula that expresses the Trinitarian reality of baptism––to be made sons and daughter of the Father, brothers and sisters of Christ, by the gift of the Holy Spirit, is that of Mt. 28:19. It acknowledge that our salvation comes from the Trinity and leads us back to the Trinity.