Catholics and Orthodox believe that the Eucharist is a sacrifice. The elements become the body and blood of Jesus and the church offers Jesus to the Father along with her other offerings and her whole self. For this reason, an ordained priesthood is important, there's an altar, there's a tabernacle to hold the elements which are not used, and other features which go along with sacerdotalism.
A virtue of this view is that it is very much in line with OT worship. In OT worship, a sacrifice was brought to the altar, a priest would offer it up, and the worshipper would consume some of the sacrifice (though this was not the case in every kind of sacrifice - the burnt offering was totally consumed in the fire of the altar). If Christianity is a fulfillment of Judaism then this makes some sense. Jesus is our sacrifice, our Passover lamb - the NT makes this plain. Also we are to consume his body and blood which was offered up as a propitiation for our sins. This would have made a lot of sense to a Jewish Christian in the first century.
However a problem with this view is that it does not appear to have New Testament warrant. I can't think of any place in the New Testament where the idea of a Eucharistic sacrifice is really made clear. I can think of many opportunities where it could have been made clear, though. But in these instances, the Eucharist is not mentioned. One would think, for instance, of the many times in the book of Hebrews when something like a Eucharistic sacrifice could have been emphasized. But this idea is absent from Hebrews.
Another problem is that the idea of a Eucharistic sacrifice appears to be missing from the literature of the apostolic fathers. This idea isn't really solidified until the third century, as far as I can tell.
Jesus is our sacrifice. Jesus is our high priest. Through Jesus we have access to the Father. But should the earthly celebration of the Eucharist be regarded as a sacrifice?
