The verse found in Deuteronomy 8:10, “You shall eat, be satiated and bless God,” is the source of the biblical mandate to recite the Birkat Hamazon (Grace After Meals, also known in Yiddish as Bentching). The Midrash, however, maintains that the custom goes back to the patriarch Abraham, who insisted that his guests give proper thanks for the meal that he had served them. This was also the perfect opportunity for Abraham to introduce the guests to the concept of one God.
The act of inviting another to recite Birkat Hamazon continues to this day in the zimmun (invitation) that precedes the blessing when three or more people eat together. It says in the Talmud, “If three people have eaten together, it is their duty to invite one another [to say Grace]” (Brachot 45a).
Just as the recitation of Birkat Hamazon is specific to a meal at which one eats bread, so too are the rules of zimmun. (In some cases, however, a m’zuman may be made up of one who has eaten bread and two who have only eaten a m'zo'note--grain based–food.)
If three people or more have begun a meal together, they should remain together until after Birkat Hamazon. The sages note that this is “because from the outset of the meal they laid upon themselves the duty to invite one another” (ibid).
The formula of zimmun consists of a formal call and response (please see text below) recited prior to Birkat Hamazon. It is customary for a host to honor one of the guests by asking the guest to lead Birkat Hamazon. It should be noted that when a minyan (quorum of 10) is present, God’s name is added to the zimmun formula. Additionally, the text of the zimmun changes in special circumstances such as the feast of a Brit Milah (circumcision), Sheva Brachot (meal following a wedding) or in a house of mourning.