According to Jewish to Jewish Tradition, Elijah the Prophet is present at every bris to help the Mohel ensure that everything goes smoothly and as planned.
Before the baby is placed for the actual circumcised, he is placed in "Elijah's Chair", sitting on the Prophet's lap for a blessing.
Unlike passover, at the bris, Elijah does not partake in the wine!
For more info about the ceremony, visit https://www.familymohel.com/the-ceremony