We read in the Mishna that we perform circumcisions on the eighth day of a child's life, regardless whether it is a weekday, Shabbat, a holiday, or even Yom Kippur. If, however, the circumcision was postponed – for health or other considerations – then the brit cannot be held on Shabbat or a Jewish holiday when "work" is prohibited.
There are two exceptions to this rule, when a brit cannot be scheduled for Shabbat: if the child was born via c-section, or if he was born between dusk and nightfall, the halachic "gray area" known as "Bein Hashmashot" (see Days for more on this topic).
Although the circumcision is performed on Shabbat, the preparations for the brit (such as bringing the implements to the brit's location or preparing bandages) must be done before the onset of Shabbat.
Normally, causing a blood-letting wound, even for a productive purpose, is forbidden on Shabbat. Why is circumcision an exception?
(The Talmudic sages debate the biblical source for the obligation to circumcise on the eighth day—even if it falls on Shabbat. Technical aspects aside, though, why does the Torah make this exception?)
There are several angles we can take to answer this question:
a) On select occasions we find that the observance of various mitzvot supersedes others. Examples: Sacrifices were offered in the Temple on Shabbat; the mitzvah of tzitzit supersedes the prohibition against wearing shatnez; practically all mitzvot are suspended in order to do the mitzvah of saving a life. The One who commanded the mitzvot told us that in certain instances, when certain mitzvot "clash," one mitzvah takes precedence over another. Performing a circumcision on Shabbat can be viewed in the same light.
b) "The children of Israel shall observe the Shabbat, to make the Shabbat throughout their generations as an everlasting covenant." The purpose of sanctifying the Shabbat, as well as abstaining from work on this holy day, is in order to maintain the everlasting covenant between G‑d and His people. As such, the brit, which is the ultimate expression of G‑d's covenant – etched in our very bodies – does not pose a contradiction to Shabbat observance. According to this reasoning, circumcision doesn't supersede Shabbat, rather it is completely in line with the spirit of the Day of Rest. We honor the Shabbat by conducting circumcisions.
c) The covenant of circumcision symbolizes our essential bond with G‑d. This essential bond transcends the Torah and mitzvot and their observance (which is why a "sinful" Jew is just as Jewish as his most righteous counterpart). Accordingly, circumcision doesn't supersede or complement the Shabbat—it exists on a level that transcends all the do's and don'ts of the Torah.
Rabbi Naftali Silberberg,
Chabad.org Editorial team