The Wikipedia article on the subject is very thorough.
Most Christians practise infant baptism. Denominations that practise infant baptism include the Roman Catholic Church, the Eastern Orthodox Church, Oriental Orthodoxy, Armenian Apostolic Church, Assyrian Church of the East, the Anglican churches, Lutherans, Presbyterians, Methodists, some Church of the Nazarene, the Reformed Church in America, the United Church of Canada, the United Church of Christ (UCC), and the Continental Reformed.
Some of the reasons for doing it include:
- Baptism of infants is extremely old, going back to at least the 1st Century and possibly New Testament times;
- Baptism is not merely symbolic, but actually a means of conveying the grace of God to the infant;
- Baptism is analogous to circumcision, which God mandated to be done to infants
- From the Book of Acts: "Peter replied, "Repent and be baptised, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit. The promise is for you and your children and for all who are far off–for all whom the Lord our God will call." (Acts 2:38-39, NIV)"
- Children are no less members of the church than adults, and should be welcomed into full membership.
Again, Wikipedia is helpful here.
It should be noted that allmany traditions that practice infant baptism have a secondary initiation, such as Confirmation, in which the person does get to "choose for themselves".