The ceremony of circumcision, by which a boy enters the Covenant of Israel on his eighth day of life, goes back to Abraham, who was commanded by God to circumcisize himself, his son Ishmael, and all the males in his household, as a sign of the covenant (Genesis, chapter 17). Since then, Jews have circumcized their sons, often risking grave danger, to welcome them into the Covenant.
Circumcision has played an important role in Jewish identity and culture for more than three millennia, and continues to do so today. Circumcision is performed on the eighth day unless health considerations advise against it (in which case it is postponed until a physician gives permission) and may even be performed on Shabbat or Yom Kippur, indicating its importance in Jewish tradition.
While parents are commanded to circumcize their sons, few are qualified to perform this surgical procedure and appoint a mohel as their agent. It is traditional to schedule the brit milah as early in the day as possible, signifying one's eagerness to observe the mitzvah. The baby is brought into the room by the sandek (godparent) who sits and hold the child in a chair that has been designated as Elijah's Chair. There are two reasons for the custom of reserving a chair as Elijah's Chair. The first, our sages related, is because one of Elijah's complaints against the Hebrews was that they had ceased circumcizing their children; hence Jewish parents demonstrate to Elijah that they are fulfilling the covenant. The second reason is because tradition teaches that Elijah will return to earth to herald the coming of the messiah. When a baby is born, it is always possible that this child is the messiah, and so he is welcomed by being held in Elijah's Chair. Following the circumcision, the child's Hebrew name is announced. The connection between circumcision and naming also derives from the seventeenth chapter of Genesis, because after God forges the covenant of circumcision with Avram, God changes Avram's name to Avraham. Ashkenazic Jews generally name their children after deceased relatives. Sephardic Jews often name their children after living relatives.
Traditionally, the only rite recognizing the birth of a daughter is naming. Her father would be called for an aliyah following her birth and prayers were said for the child's and mother's health, and her name was announced publicly.
Over the past two decades, ceremonies have been developed to welcome girls into the Jewish covenant, filling a void in the tradition. There are a variety of ceremonies in existence, but none has been universally accepted as yet. Brit Rechitzah involves washing the child's feet as a sign of welcome, based on the account in Genesis in which Abraham washes the feet of the strangers (angels) as a gesture of welcome and hospitality. In some ceremonies, candles are lit. Prayers and blessings are recited and the child's Hebrew name is announced.
In B'Midbar (the Book of Numbers 18:15-16) we read: "The first issue of the womb of every being, human or animal, that is offered to the Lord shall be yours [the priest's], but the firstborn of humans shall be redeemed, and the firstlings of unclean animals shall be redeemed. Take as their redemption price from the age of one month up, the money equivalent to five shekels by the sanctuary weight, which is twenty gerahs." The Torah claims for God every firstborn, human and animal. The firstborn male of ancient Israelite families had special obligations since, from the day of birth, he was consecrated to the vocation of assisting the priests in the conduct of the sacrificial cult. Once the Tabernacle was built, the duty was transferred to the Levites. Since that time, firstborn Jewish males have been released from their obligation through a ceremony called Pidyon HaBen. It is not widely practiced by liberal Jews these days.
Pidyon HaBen applies only to the firstborn male child of a Jewish woman who "opens the womb," that is, who is delivered vaginally. Babies delivered by Caesarean-section do not have to be redeemed because they did not "open the womb". In addition, if the woman had previous been pregnant but miscarried, redemption of a subsequent full term pregnancy is required only if the miscarriage took place within 40 days of conception. If a woman's first child was delivered by C-section, and she subsequently gives birth to a son, the second born son does not have to be redeemed (because he is not firstborn). Finally, the firstborn of Kohanim and Leviim are exempt, as are the firstborn of daughters of Kohanim and Leviim.
The redemption ceremony takes place when the child is a full thirty days old, hence from the thirty-first day of life, unless that day falls on a Shabbat or festival, in which case it is postponed one day because the ceremony involves a monetary transaction. It is customary to use five silver dollars, since we no longer have shekalim and the dollars are considered equivalent for this purpose. The five silver dollars are transferred to a Kohan during the ceremony, and are often donated to charity.
Our history is replete with examples of adoption. Moses was raised by an adoptive mother, the daughter of the pharaoh of Egypt. The midrash tells us that she loved him dearly, and hugged and kissed him and protected him constantly. Esther, who later became Queen of Persia, was raised by her uncle Mordecai, who adopted her at a young age. The Talmud (Sanhedrin 19b) expounds upon these cases noting that, "This teaches you that whoever brings up an orphan in his home, Scripture imputes to him as though he had begotten him."
The different streams of Judaism deal with questions arising from adoption differently, and so it is best to consult your rabbi with regard to the use of the parents' Hebrew names as part of the child's name, circumcision for boys, mikveh for both boys and girls (if the child needs to be converted), etc. There are organizations of Jewish parents who have adopted cross-culturally which can provide information and support.
The New Jewish Baby Book by Anita Diamant (1993, Jewish Lights Publishing).
The Complete Dictionary of English and Hebrew First Names by Alfred Kolatch (Jonathan David Publishing).
The Brit Milah Board of the Reform Movement