The night before an infant’s Berit Mila is called "Berit Yishak" ("The Covenant of Yishak"), or in Arabic, "Shad Elias" ("The Angel Eliyahu"). We refer to this night with these names because Yishak was the first person to have a Berit Mila on the eighth day (Abraham was the first to have a Berit, but he was 99 years old), and Eliyahu attends every Berit.
At the Berit Yishak, ten men are assembled and they recite a portion of text from Zohar. It is proper for Torah scholars to be present at this event. The custom of Berit Yishak is based upon a passage in the Zohar (Parashat Lech-Lecha) that speaks of the importance of having Hachamim studying Torah in honor of the child on the night before the Berit. The Zohar tells the story of Rabbi Abba and his son, Rabbi Yaakob, who arrived at a certain place of lodging and the woman of the house asked them to stay because her son was undergoing Berit Mila the next day. The two Sages agreed, and they stayed awake the entire night studying Torah. Similarly, the Hid"a (Rav Haim Yosef David Azulai, 1724-1807), in his Shem Ha’gedolim (listen to audio recording for precise citation), relates that a certain great Rabbi once took gravely ill, until Eliyahu Ha’nabi appeared to him. The prophet told him that he should take it upon himself to go to the home of every newborn boy the night before the Berit to learn for the child. The Rabbi committed himself to observe this practice, whereupon he was instantly and miraculously cured. From that time on, he fulfilled his commitment and went to learn Torah at the home of every newborn boy before the Berit.
The reason for this custom is that the Mazikin (harmful spiritual forces) try to cause harm to the child and prevent him from having a Berit Mila, and it is therefore necessary to grant the child protection through the study of Torah. This is alluded to in God’s formulation when commanding Abraham about the Berit Mila: "Ve’ata Et Beriti Tishmor" ("And you shall observe My covenant"), indicating that the Berit requires "Shemira" – protection. This protection is provided through the study of Torah the night before the Berit.
The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) records a custom to light candles in honor of Eliyahu Ha’nabi at the Berit Yishak, and to place coins for charity on the plate where the candles are lit. Our custom is to light the candles at the Berit itself, and not the night before.
It is customary at the Berit Yishak for the mother to come to the side room and recite Birkat Ha’gomel. Since ten men are present, this event provides the woman with the opportunity to recite Birkat Ha’gomel for having survived childbirth, which is considered a dangerous condition. Some scholars claim that this is also the reason for the custom to recite the verse, "Hodu L’Hashem Ki Tob Ki Le’olam Hasdo" at the Berit Mila, to give praise and thanks to the Almighty for bringing the woman safely through childbirth. (Others say that if this were the case, this verse should be recited even after the birth of a girl, and there would be no reason to recite it specifically at the Berit Mila. They thus explain that at the Berit, a newborn boy’s soul is freed from the state of confinement in which it was held during the first week, and we recite the verse of "Hodu L’Hashem" to celebrate the soul’s release from its confinement.)
Summary: It is customary on the night before a Berit Mila to assemble ten men in the home, as well as Torah scholars, to read special passages from the Zohar, in order to give the baby protection. During this event the woman should come to the side room and recite Birkat Ha’gomel.