(Luke 2:21-34, 38)
There were several ceremonies that had to be performed after a baby's birth: circumcision, redemption of the firstborn, and the purification of the mother. Eight days after birth, every boy was circumcised and named (Leviticus 12:3; Luke 1:59,60). Circumcision was a symbol of the Jewish peoples' unique relationship with God and a sign of the Jews' separation from the Gentiles. This was an extremely important ceremony to the family of a Jewish baby boy. Circumcision was commanded by God when he was beginning to form his holy nation (Genesis 17:4-14), and was reaffirmed through Moses (Leviticus 12:1-3).

Redemption of the firstborn was a ceremony signifying that the parents of the baby boy acknowledged the child belonged to God, who alone has the power to give life. This ceremony was to take place one month after birth (Exodus 13:2, 11-16; Numbers 18:15, 16). It also included buying back--"redeeming"--the child from God through an offering.

The third ceremony that had to take place was the purification of the mother. The mother was considered ceremonially unclean and could not enter the temple for 40 days after the birth of a son and 80 days after the birth of a daughter. At the end of her time of separation, the parents were to bring a lamb for a burnt offering and a dove or pigeon for a sin offering. If they could not afford to bring a lamb, they were allowed to bring a second dove or pigeon.
(Life Application Bible, New International Version, Tyndale House Publishers, Inc., Wheaton Illinois, and Zondervan Publishing House, Grand Rapids, Michigan, 1991, 1792.)

Hebrew Law and Custom

The Law of Moses

Luke 2:21-24.
The requirements relevant for Mary and Joseph as new parents are explained in Leviticus 12 (circumcision for baby Jesus and purification for Mary), and Exodus 13:2 and 12 (devoting the first-born to the Lord).