The Shulchan Aruch[2] rules that a woman may perform a brit mila, although it is preferable that an adult male do so. The Rama (which is the primary source for Ashkenazic practical halacha) cites an opinion that holds that women may not perform the brit, and he states that the custom follows this position.
This disagreement stems from the gemara,[3] which mentions varied views of Tanna’im regarding whether a brit mila done by a non-Jew is valid. Rav and Rabbi Yochanan argue concerning the reason why a non-Jew’s brit mila would be disqualified. Rav said it is because they do not belong to the group that was specifically commanded to have a brit. Rabbi Yochanan said that only one who is circumcised may circumcise others.
The gemara points out that, according to Rav, a woman is disqualified because brit mila doesn’t apply to her. However, according to Rabbi Yochanan, a Jewish woman, as a member of the covenantal nation, is considered as if she is circumcised and, therefore, is qualified. Interestingly, a circumcised non-Jew is considered uncircumcised because he is not part of the covenant of which the brit mila is a sign. (Note that “brit” means covenant.) The Rishonim disagree as to which opinion should be accepted as halacha, as do the aforementioned Shulchan Aruch and Rama.
We conclude with the gemara’s attempted proof that a woman may perform a brit mila. After all, Tziporah circumcised Eliezer to save Moshe’s life.[4] The gemara, however, deflects the proof by noting that it is possible that she gave the rock to someone else to do the brit or that she only started the mila, but Moshe completed the job.