Answer:

Since in my church, we do baptize babies, and consider it a "Sacrament", first, I will address the issue of what a Sacrament actually is...

A Sacrament is an outward sign, instituted by Christ, to give inward grace. There are seven Sacraments: Baptism, Confirmation, Penance, Holy Eucharist, Anointing of the Sick, Holy Orders, and Matrimony. The Church uses numerous ceremonies or actions in applying the outward signs of the Sacraments to increase our reverence and devotion for the Sacraments, and to explain their meaning and effects. All the Sacraments were instituted by Our Lord, for God alone has power to attach the gift of grace to the use of an outward or visible sign. The Church, however, can institute the ceremonies to be used in administering or giving the Sacraments.

The Sacraments have been instituted as a special means through which we are to receive the grace merited for us by Christ. As Christ is the giver of the grace, He has the right to determine the manner in which it shall be given, and one who refuses to make use of the Sacraments will not receive God's grace.

The Sacraments recall in many ways the means by which Our Lord merited the graces we receive through them. Baptism recalls His profound humility; Confirmation His ceaseless prayer; Holy Eucharist His care of the needy; Penance His mortified life; Extreme Unction His model death; Holy Orders His establishment of the priesthood, and Matrimony His close union with the Church.

The soul is brought into spiritual life by Baptism; it is strengthened by Confirmation; nourished by the Holy Eucharist; healed by Penance; helped at the hour of our death by Extreme Unction; guided by God's ministers through the Sacrament of Holy Orders, and it is given a body in which to dwell by the Sacrament of Matrimony. Some of the Sacraments give sanctifying grace, and others increase it in our souls.

Since the early Christian era, the Catholic Church has always understood baptism as a Sacrament which accomplishes several things, the first of which is the remission of sin, both original sin and actual sin (only original sin in the case of infants and young children, since they are incapable of actual sin; and both original and actual sin in the case of older persons). It in no way acts to replace or somehow "repair" the Redemption that Christ gave us.

Infant baptism is an initiation of that child into the Church... not only the local, actual church that we attend, but of the "universal" Church. That is a beautiful thing to think about! The Catholic Church is a church that is basically the same all around the world... the same beliefs, the same traditions, the same Lord (maybe just different languages and styles of worship). This child is to be supported and loved and taught by all of its new brothers and sisters in Christ in the church.

It is also a true call to conversion to the parents. I wonder sometimes if it's of more importance for the parents than for the babies. When parents (and godparents) bring their baby before the church to ask for baptism for her, they are saying, "God, we are bringing her to You. This is the beginning of our job as our child's spiritual parents. We want to start her on her journey back to You. We're also asking these people to help us do that."

Infant baptism doesn't mean quite the same thing as baptism does in the Protestant church. When a child is born, it is born into a spiritual poverty (which some would call original sin).  Years ago some Catholics might have described it as a black spot on the soul that was inherited from our parents' (and their parents all the way back to Adam) sins that needed to be washed away in baptism. That’s what I was told as a little girl.

Baptism, as a Sacrament, is not just a symbolic action (that is such a huge difference between the Catholic Church and Protestant churches!). As a Sacrament, a real change takes place... the Holy Spirit fills that child's spirit.  Over time, because of personal sin, that child can lose the indwelling of the Holy Spirit and will need to come to the Lord later to ask for forgiveness and for Him to be his Savior (this is called Confirmation. Even though some Catholics might say they aren't "born again", this is when they really are... they make a public decision to follow Christ and make Him Lord of their lives. It's more difference in terminology. Of course, there are surely some Catholics who aren't "saved", just as there are some Baptists who aren't "saved". But that's between them and God.)

Peter explained what happens at baptism when he said, "Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit" (Acts 2:38).  But he didn't restrict this teaching only to adults. He added, "For the promise is to you and to your children and to all that are far off, every one whom the Lord our God calls to him" (2:39).

We also read: "Rise and be baptized, and wash away your sins, calling on his name" (Acts 22:16). These commands are universal, not restricted to adults.

The connection between baptism and salvation is clear, not only because of these passages, but because of other explicit passages such as 1 Peter 3:21: "Baptism . . . now saves you, not as a removal of dirt from the body but as an appeal to God for a clear conscience, through the resurrection of Jesus Christ."

Along with this forgiveness of sins, along with the Sacrament of Baptism comes an infusion of grace. It is this grace that makes the soul spiritually alive and capable of enjoying heaven. There are other benefits, too, such as the right to special graces necessary to enable the baptized to fulfill his baptismal promises.

The Catholic Church has always held that Christ's law applies to infants as well as adults. Jesus said that no one can enter heaven unless he has been born again of water and the Holy Spirit (John 3:5). His words can be taken to apply to anyone capable of having a right to his kingdom. He asserted such a right even for children: "Let the children come to me, and do not hinder them; for to such belongs the kingdom of heaven" (Matt. 19:14).

More detail is given in Luke's account of this event, which reads: "Now they were bringing even infants to him that he might touch them; and when the disciples saw it, they rebuked them. But Jesus called them to him, saying, "Let the children come to me, and do not hinder them; for to such belongs the kingdom of God" (Luke 18:15-16).

As you can see, the text in Luke 18:15 says, "Now they were bringing even infants to him", and following this are the same words as in Matt. 19:14. The word "infants" was specifically used --children who are quite unable to approach Christ on their own and who could not possibly make a conscious decision to "accept Jesus as their personal Lord and Savior."

And that is precisely the problem. Protestant doctrine don't permit the baptism of infants and young children because they are not yet capable of making such a conscious act. But notice what Jesus said: "to such as these [referring to the infants and children who had been brought to him by their mothers] belongs the kingdom of heaven."

The Lord did not require them to make a conscious decision. He says that they are precisely the kind of people who can come to him and receive the kingdom. If Jesus said "let them come unto me," who are we to say "no," and withhold baptism from them?

The Bible nowhere says baptism is to be restricted to adults. Some just conclude that's what it should be taken as meaning, even if the text doesn't explicitly support such a view. Naturally enough, the people whose baptisms we read about in Scripture (and there aren't many who are individually identified) are adults because they were converted as adults. This makes sense, because Christianity was just starting out and there were no "cradle Christians," no people brought up from childhood in Christian homes.

But, one might ask, does the Bible ever say that infants or young children can be baptized? The indications are fairly clear. Lydia was converted by Paul's preaching. "She was baptized, with her household" (Acts 16:15). The Philippian jailer whom Paul and Silas had converted to the faith, who had been about to commit suicide when they were miraculously freed from their imprisonment, was baptized that night along with his household. We are told that "the same hour of the night . . . he was baptized, with all his family" (Acts 16:33). And in his greetings to the Corinthians, Paul recalled that, "I did baptize also the household of Stephanas" (1 Cor. 1:16).

In all these cases, whole households or families were baptized. This means more than just the spouse; the children too were included. If the text of Acts referred simply to the Philippian jailer and his wife, then we would read that "he and his wife were baptized," but we don't. Thus his children must have been baptized as well. The same applies to the other cases of household baptism in Scripture.

Granted, we can't tell the age of the children; they may have been past the age of reason, rather than infants. Then again, they could have been babes in arms. More probably, there were both younger and older children. Certainly there were children younger than the age of reason in some of the households that were baptized. Furthermore, if given the New Testament pattern of household baptism, if there were to be exceptions to this rule (such as infants), they would be spelled out.

Infant baptism accords perfectly with early Christian practices. Origen, for instance, wrote in the third century that "The Church received from the apostles the tradition of giving baptism also to infants." Augustine said of infant baptism: "This the Church always had, always held; this she received from the faith of our ancestors; this she perseveringly guards even to the end." Cyprian wrote that "From baptism and from grace . . . must not be kept the infant." The Council of Carthage, in 253 AD condemned the opinion that infants should be withheld from baptism until the eighth day after birth.

There is a good essay on the early Church's writings on infant baptism, with quotes from various early Christian writers.

http://www.catholic.com/ANSWERS/tracts/_infant.htm