Long before most people arrive at Church, the priest prepares the Holy Gifts (the bread and wine that will be consecrated into the body and blood of Christ) in a small service called the Proskomedi. He cuts pieces of a loaf of bread called the prosphora (“before the gifts”) and places them on dish called a paten, and then pours water and wine into the chalice. This is what is carried through the congregation during the Great Procession in the Divine Liturgy. This is also the time when the names you submit to the priest are first read.
Fr. John Peck, pastor of St. George Greek Orthodox Church in Prescott Arizona prepared a video on the Proskomedi Service. You can view it here.