The baby is brought in to the room, carried by the kvatter and kvatterin, considered by many to be the godparents.
Two chairs are prepared for the Brit. The first is for the Sandek, the individual who holds the baby on their knees during the actual circumcision. The lap of the Sandek is considered analogous to the altar of the Temple itself. It is considered a great honour to be the Sandek because there is a Kabbalistic tradition that links the soul of the Sandek with the child. In this way, the Sandek is considered the spiritual mentor of the child. In many instances, one of the grandfathers serves as the Sandek.
The second chair is set aside for the spirit of Elijah the Prophet, the "Angel of the Covenant". According to Jewish tradition, Elijah comes to every circumcision to testify before the Almighty to the commitment of the Jewish people to this great mitzvah throughout the generations. During the ceremony, just prior to the Brit itself, the baby is placed on the chair of Elijah, and the Mohel recites a special prayer asking for the spirit of Elijah to stand over him as he performs the Brit.
After the Mohel has performed the brit, a special blessing is recited upon a cup of wine, and the baby is given his Hebrew name. It is customary to serve refreshments or a meal after the brit, and this is considered a seudat mitzvah, part of the mitzvah. (Obviously all food served should be kosher.)
Ideally, a minyan should be present for a Brit, although this is not a pre-requisite.