Tradition teaches that Eliyahu Ha'navi is present – in one form or another – at every Berit Mila, and for this reason a chair must be designated for him. One should expressly designate the chair by saying, "Zeh Kisei Shel Eliyahu Ha'navi Zachur Latov" ("This is the chair of Eliyahu Ha'navi"). (See TUR, siman 265, and Shulchan Aruch Yore Deah, siman 265:11)
Why does Eliyahu attend every Berit?
The Gemara tells that Eliyahu, who lived during the reign of the idolatrous king Achav, when many Jews abandoned the Torah, came to God and complained that Benei Yisrael were not observing the Mitzva of circumcision. God reprimanded Eliyahu, saying that He does not need a prophet to prosecute against His people, to accuse them of disloyalty. The prophet's job is to pray on their behalf and show them compassion even if they sin. God therefore decreed upon Eliyahu that for all time he must attend every circumcision ritual performed by Benei Yisrael, and then return to God and report that the Jewish people faithfully observe this Mitzva.
It is further told that Eliyahu then told God that he would be unable to tolerate attending a Berit Mila if the child's father is a sinner. After all, Eliyahu was a zealot, who could not tolerate any sin or infringement upon the Almighty's honor. (The Sages teach that Eliyahu was Pinchas, who, as we know from the Torah, was a zealot when it came to God's honor.) God replied that He will forgive all the father's sins. Eliyahu then noted that the Mohel, or perhaps one of the guests, might have a sinful record, and God again responded that He will forgive the sins committed by the Mohel and all the attendees. (See Taamei HaMinhagim)
For this reason, attending a Berit Mila is considered an effective means of achieving atonement. It should be noted that according to the Benei Yissaschar (work by the Chassidic sage Rabbi Tzvi Elimelech of Dinov, 1783-1841), attending a Berit earns one atonement only if he stands near Eliyahu's seat. Given Eliyahu's zealotry, those near him at the Berit Mila require atonement lest he react angrily to their sins. One should therefore endeavor to stand near Eliyahu's chair at a Berit Mila. Additionally, some Rabbis write that one earns atonement at a Berit Mila only if he performs Teshuva. Attending a Berit does not obviate the need for repentance, but rather allows a person to repent and earn atonement more easily. It is therefore appropriate when attending a Berit Mila to contemplate sincere thoughts of remorse and repentance.
Summary: At a Berit Mila a chair must be verbally designated for Eliyahu Ha'navi, who, as tradition teaches, attends every Berit Mila. Attending a Berit Mila has the capacity to earn one atonement, particularly if he stands near Eliyahu Ha'navi's chair, and therefore guests at a Berit should contemplate thoughts of Teshuva and stand near Eliyahu Ha'navi's chair.
(See Soba Semahot, Helek 2m page 66)