A baby boy needs to be redeemed if he is the firstborn and he was born to Israelite parents. Let's look at what each of these means in detail.

Firstborn

The Torah's term for the firstborn is "the opener of the womb." This means that it is the mother's firstborn baby, if it is male, who is required to have a pidyon haben (redemption ceremony)—even if the father already has other children.

If a daughter is born first, then no redemption of a subsequent male child is necessary.

If the mother's first pregnancy ended in stillbirth, the subsequent child does not have a pidyon haben. If she miscarried within the first forty days of gestation, and a son is born next, he would need to have a pidyon haben. If she miscarried after the first forty days, a rabbi should be consulted whether the next child is considered "the opener of the womb."

The obligation only applies if both of the parents are IsraelitesIf the firstborn child was delivered via caesarean section, then no pidyon haben is held—not for the firstborn, and not for the next child, even if the next child was delivered naturally.

If twin boys are born, only the firstborn must be redeemed. If a boy and a girl are born, the boy must only be redeemed if he is born first.

Israelite

The obligation only applies if both of the parents are Israelites. If either the father or the mother is the child of a father who is a kohen (priest) or Levite, the pidyon haben is not required.1

If a woman converts to Judaism (even whilst pregnant) her firstborn requires a pidyon haben. If a women who has already had children converts, her first Jewish-born son does not need to be redeemed.

Who redeems the child?

The obligation to redeem the firstborn son rests upon the father. The mother has no responsibility to arrange for her son's redemption. The obligation on the father kicks in when the child reaches thirty days of age, and – in the event that the pidyon haben was not arranged in its proper time – continues until the child's bar mitzvah. Once the child has reached the age of adulthood, the mitzvah transfers to him, and he is required to redeem himself from a kohen. (A rabbi should be consulted for the exact procedure for "self-redemption.")

If the father is unavailable to redeem his son for whatever reason – e.g., he is deceased or is not Jewish – technically no one is obligated to redeem the child until his bar mitzvah, at which point the child is required to redeem himself. Nevertheless, the mother, a grandfather, or even the local Jewish community can redeem the child. (A rabbi should also be consulted in this situation, too, to advise as to the procedure of a "fatherless redemption.")