Hebrew Bible
The practice is referred to in the Gospel of Luke 2:22-24.

New Testament
When a Jewish woman gives birth by natural means, i.e. vaginally (and not by Caesarean section) to her firstborn if it is a son then the child must be "redeemed".

Principles
In the traditional ceremony, the father brings the child to the Kohen and recites a formula, or responds to ritual questions, indicating that this is the mother's firstborn son and he has come to redeem him as commanded in the Torah. The Kohen asks the father which he would rather have, the child or the five silver shekels which he must pay. The father states that he prefers the child to the money, then he recites a blessing and hands over five silver coins (or an equivalent amount of total silver). The Kohen holds the coins over the child and declares that the redemption price is received and accepted in place in the child. He then blesses the child and returns him to the custody of his family.
The ceremony traditionally takes place amidst a minyan of 10 men. The child is sometimes presented on a silver tray, surrounded by jewelry lent for the occasion by women in attendance. The event is accompanied by a meal, and guests in some places are given cloves of garlic and cubes of sugar to take home: these strongly-flavored foods can be used to flavor a large quantity of food which will in some sense extend the mitzvah of participation in the ceremony to all who eat them.

Ceremony
Contemporary religious authorities believe that the Shekel HaKodesh (Holy Shekel) of the Temple was larger and of purer silver content than the standard Shekel used for trade in ancient Israel. Halakha requires that the coins used have a requisite total amount of actual silver, which contemporary religious authorities believe to be approximately 117 grams. Coins which do not contain the requisite amount of silver do not result in a valid redemption.
The Israeli Mint has minted special edition 23.4 gram silver commemorative coins for the purpose, five of which would come to exactly 117 grams of silver. Pre-1965 American silver dollars weigh 26.73 grams of 90% silver content and hence contain 24.06g of pure silver, although such coins have become increasingly rare (modern U.S. coins contain no silver). Four American Silver Eagle coins, specially minted coins sold to collectors and investors which contain 31.103 grams of 99.9% pure silver, or five of the above-mentioned specially minted silver coins of Israel are commonly used for Pidyon Ha-Ben in the United States.
Though the silver coins are the payment to the Kohen under Jewish law, they are usually returned to the family as a gift for the child, as the coins themselves are often commemorative in nature. There are many examples of artistically crafted gift boxes or display cases made for the child to have as a memento of the occasion. The father then usually offers a gift or fee of more conventional cash to the Kohen.
Some Kohanim sell coins of sufficient weight and purity of silver to facilitate the ceremony, as such coins are usually not readily obtainable.

Coins
Some Orthodox authorities, citing a passage in the Talmud (Kiddushin 8a) describing such an event, permit a male non-Kohen married to a Bat Kohen (daughter of a male Kohen) to accept Pidyon HaBen money on the Bat Kohen's behalf.. The question of a Bat Kohen accepting Pidyon HaBen money on her own behalf is a matter of discussion in Modern Orthodox Judaism but is not currently done in practice. No branch of Judaism currently accepts a Pidyon HaBat (redemption of a first-born daughter) ceremony.

Women and Pidyon HaBen
Conservative Judaism requires a Pidyon HaBen ceremony under the same circumstances as Orthodox Judaism. The Committee on Jewish Law and Standards (CJLS) permits a Bat Kohen (daughter of a male Kohen) to perform the Pidyon Haben ceremony on her own behalf.

In Conservative Judaism
Consistent with their views that Temple- and priesthood-related rituals and statuses are archaic and inconsistent with modern egalitarian values, Reform Judaism and Reconstructionist Judaism generally do not perform the Pidyon HaBen ceremony.

In Reform and Reconstructionist Judaism
Pidyon HaBen is one of the rarest Jewish ceremonies usually applying to approximately 1 in 50 Jewish births . The firstborn must be a boy, caesarian section births are not eligible, a previous miscarriage precludes the ceremony and neither parent must be a kohen or a Levi. The ceremony is even rarer because it generally not performed in Reform Judaism and Reconstructionist Judaism, which have generally abolished the status of Kohen and special traditional ceremonies involving them.

Frequency of ceremony
According to the traditional rabbinic interpretation, in the early part of the Bible, as recorded in the Book of Genesis, the duties of a priest fell upon the eldest son of each family. The first-born was to be dedicated to God in order to perform this task.
Following the Israelite Exodus from Egypt, after the nation had sinned with the Golden Calf, the priesthood was taken away from the first-borns, and given to the tribe of Levites, specifically to the Kohenim, High Priest Aaron, his children, and their descendants. At the same time it was instituted that the first born of each family should be redeemed; i.e. they would be 'bought back' from the dedication to God that would previously have been required of them. Levites were substituted for the first-born and wholly given to Divine service:
And thou shalt give the Levites unto Aaron and to his sons; they are wholly given unto him from the children of Israel.
And I behold, I have taken the Levites from among the children of Israel instead of every first-born that openeth the womb among the children of Israel; and the Levites shall be Mine. For all the first-born are Mine: on the day that I smote all the first-born in the land of Egypt I hallowed unto Me all the first-born in Israel, both man and beast, Mine they shall be: I am the LORD.' (Numbers 3:9, 12-13)
The first-born male of every clean animal was to be given up to the priest for sacrifice (Deuteronomy 12:6; Exodus 13:12, 34:20; Numbers 18:15-17). The first-born of unclean animals, however, was either to be redeemed or sold and the price given to the priest (Leviticus 27:11-13, 27). The first-born of an ass, if not redeemed, was to be put to death (Exodus 13:13; 34:20).

Traditional Jewish interpretation
Biblical criticism perspectives are based on the Documentary Hypothesis, that the Torah was compiled by an editor from four different sources, each with a different perspective.
According to Peake's Commentary on the Bible, the passages in the Hebrew Bible referring to the redemption of the firstborn as reflecting traditions that already existed, rather than being the origin of them.

See also

This entry incorporates text from the public domain Easton's Bible Dictionary, originally published in 1897.