There are 613 mitzvot that Jews are supposed to keep. Some are positive and some negative. The positive ones equal 248, which also equals the number of bones and vital organs in our body. There are 365 negative commandments, which is also the number of days in a solar year. Some mitzvot we can no longer do, as we do not have the Bet haMikdash, Holy Temple. Others are only required for those living in Israel. Some, like tzitzit, are only for men and others, like candle lighting, only for women. Welcoming a baby boy into the Covenant of Abraham is one time-honored mitzvah.
In Genesis, chapter 17, there are several lines where G-d explains to our patriarch Abraham the rules on circumcision. The commandment was not only meant for Abraham but for all the men in his household to be circumcised. In Leviticus 12:3 there is yet another reference on how a male child of eight days old must have a Brit Milah. This mitzvah binds the newborn child to his parents and to Hashem (G-d). As Abraham was responsible for making sure that his sons and household members were circumcised, it is necessary for a father to have his son brought into the Covenant.
Abram was born into an idol-worshipping family, and when he decided that there was only one true G-d, his name was changed to Abraham. His wife, Sarai, became Sarah. The name change is also symbolic as after the Brit Milah (in Orthodoxy), the infant boy is given his name. A baby girl is named in synagogue when, usually, the father is called up for an Aliyah to the Torah.
The Brit Milah is so important that even if the eighth day is on Shabbat, when the sanctity of the day often overrules some commandments, the ceremony takes place. If the child is born by Caesarian, which could mean the child may have been born on any other day naturally, the Brit takes place on Sunday. Judaism is always concerned with health, so if the child is not well enough to have this surgical procedure it is postponed until the infant is healthy enough. The mohel, the person trained to do the Brit Milah, and a doctor often consult on these issues.