No. 1282, 83 and 84, page 325 of the Catechism of the Catholic Church: The Celebration of the Christian Mystery; Libreria Editrice Vaticana: Urbi Et Orbi Communications:
Since the earliest times, Baptism has been administered to children, for it is a grace and a gift of God that does not presuppose any human merit; children are baptized in the faith of the Church. Entry into Christian life gives access to true freedom.
With respect to children who have died without Baptism, the liturgy of the Church invites us to trust in God's mercy and to pray for their salvation.
In case of necessity, any person can baptize provided that he have the intention of doing that which the Church does and provided that he pours water on the candidate's head while saying: "I baptize you in the name of the Father, and of the Son, and of the Holy Spirit."
Q. Is it wrong to defer the baptism of an infant?
A. It is wrong to defer the baptism of an infant, because we thereby expose the child to the danger of dying without the Sacrament.
Q. Why is the name of a saint given in Baptism?
A. The name of a saint is given in Baptism in order that the person baptized may imitate his virtues and have him for a protector as patron saint.
Q. Where will persons go who - such as infants - have not committed actual sin and who, through no fault of theirs, die without baptism?
A. Persons, such as infants, who have not committed actual sin and who, through no fault of theirs, die without baptism, cannot enter Heaven; but it is the common belief they will go to some place similar to Limbo, where they will be free from suffering, though deprived of the happiness of Heaven.
Q. Can a person who has not himself been baptized, and who does not even believe in the Sacrament of baptism give it validly to another in case of necessity?
A. A person who has not himself been baptized, and who does not even believe in the Sacrament of baptism, can give it validly to another in case of necessity, provided: (1) He has the use of reason; (2) Knows how to give baptism, and (3) Intends to do what the Church intends in giving of the Sacrament. Baptism is so necessary that God affords every opportunity for its reception.