The obligation to circumcise a son is found in parshas Lech Lecha, when God tells Abraham that every male among his descendants must have the foreskin of the penis removed on the eighth day. (This mitzvah was reiterated in parshas Tazria, Leviticus 12:3). By performing this act, we partner with God in completing the act of creation.
The reason for this mitzvah is that it forms a permanent, physical sign of our covenant with God. The Talmud in Menachos 43b relates that King David was saddened when he entered the bath house, since he was bereft of his tallis, tefillin and other physical symbols of the Jewish people’s commitment to fulfilling God’s will. When he saw the place of his circumcision, however, he was comforted.
This mitzvah is in force at all times and in all places. The obligation falls on the father of a baby boy and, in the absence of a father, on the local beis din. (A mother is not obligated in circumcising her son.) If the boy reaches the age of majority without being circumcised, he becomes responsible to see to it himself. Failure to perform this act, called milah, is one of the rare positive mitzvos punishable by kareis (spiritual excision). (Kareis would only apply to an uncircumcised male who willfully fails to rectify the matter.)
Milah is performed even on Shabbos, but only if that is the eighth day following a vaginal birth. (A baby delivered via Caesarian section, or one whose bris is postponed for health reasons, would not be circumcised on Shabbos.)
This mitzvah can be found in the 19th chapter of the Talmudic tractate of Shabbos, as well as in Yevamos and elsewhere. In the Shulchan Aruch, it can be found in Yoreh Deah 260. It is #215 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos and #47 of the 77 positive mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.