Just before the Mohel performs the act of circumcision, he recites a Beracha over the Misva of Berit Mila – "Baruch At Hashem…Asher Kideshanu…Ve’sivanu Al Ha’mila." The father also recites a Beracha – "Baruch Ata Hashem…Asher Kideshanu…Ve’sivanu Le’hachniso Bi’brito Shel Abraham Abinu." However, three different views exist among the Rishonim (Medieval Halachic scholars) as to when the father recites this Beracha. The Rashbam (Rabbi Shemuel Ben Meir, France, 1085-1158; grandson of Rashi), as cited by Tosafot (Shabbat 137), ruled that the father recites this Beracha at the very beginning of the procedure of Berit Mila, before the Mohel recites his Beracha and performs the circumcision. Just as all Berachot recited over Misvot are recited before the Misva act, here, too, the father should recite the Beracha before the Misva is performed. This is also the opinion of the Rif (Rav Yishak of Fez, 1013-1103), and of Rabbenu Abraham Ben Ha’Rambam (1186-1237).
At the opposite extreme, Rabbenu Tam (Rabbenu Yaakob Ben Meir, 1100-1171; brother of the Rashbam), cited by Tosafot elsewhere (Masechet Pesahim), maintained that this Beracha is recited after the conclusion of the Berit. Rabbenu Tam argued that this Beracha is not recited over the Misva of Berit Mila, as it falls under the category of "Birkot Ha’shevah" – blessings of praise and thanksgiving. The father recites this Beracha not over his personal observance of the Misva to have his son circumcised, but rather to praise G-d for the great privilege our nation has been granted to join the covenant of Abraham Abinu, becoming part of G-d’s special nation. Berachot of praise are recited after the event for which we give praise, and thus, according to Rabbenu Tam, the Beracha of "Le’hachniso" should be recited only after the Berit is performed.
A third opinion is that of the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327), who ruled that this Beracha should be recited in between the two stages of the circumcision – the "Mila" and the "Peri’a." After the Mohel cuts the foreskin, but before he pulls down the thin membrane underneath the foreskin, at that point, the father recites the Beracha of "Le’hachniso."
The Shulhan Aruch (Yoreh De’a 265:1) rules in accordance with the Rosh’s position, however, the practice among Syrian Jews does not follow the Shulhan Aruch’s ruling in this regard. Rav Eliyahu Shama Ha’levi (Chief Rabbi of Aleppo, d. 1814), in his work Machshireh Mila – which serves as the Berit Mila "handbook" for Syrian Jewish communities – writes that the custom among Syrian Jews follows the Rashbam’s position, that the father recites the Beracha of "Le’hachniso" at the beginning of the Berit Mila, before the Mohel recites his Beracha. This tradition among Syrian Jews predates the Shulhan Aruch’s ruling, and thus we do not follow the Shulhan Aruch’s position on this matter.
It should be noted that if the baby’s father is performing the circumcision, then according to all opinions, he recites the Beracha of "Le’hachniso" before beginning the Berit. Practically speaking, it would be very difficult for the father to pause in between the "Mila" and the "Peri’a" to recite the Beracha, and this would also prolong the infant’s pain, which we obviously want to avoid. And thus in this instance, according to all opinions, the Beracha of "Le’hachniso" is recited at the very beginning, before the father begins performing the Berit.
Summary: Just before performing a Berit Mila, the Mohel recites the Beracha of "Al Ha’mila." A different Beracha – "Le’hachniso Bi’brito Shel Abraham Abinu" – is recited by the baby’s father. According to the tradition followed by Syrian Jewish communities, the father recites this Beracha at the very beginning of the process, before the Mohel recites his Beracha and performs the Berit. In other communities, the father recites this Beracha immediately after the Mohel cuts the foreskin, unless the father himself serves as Mohel, in which case he recites the Beracha of "Le’hachniso" before performing the Berit.