What do Orthodox Christians mean when they talk about the Real Presence in the Eucharist?

In the Anglican Catholic Church, the change to the elements of the wafer and wine is mysterious, and no-one tries to explain what has happened. But there is no doubt that the wafer and wine in some way become the body and blood of Christ, and that Christ is indeed present during the Eucharist. Christ is also present if any surplus wafers are reserved on the altar - a light is suspended to the left of the altar to signify the presence of Christ.

This is an addendum: Anglican Catholic doctrine is also Orthodox doctrine. Our Bibles contain the same books (80). The Greek Church Fathers are accepted as authorities in the Anglican Catholic Church. The Church Fathers include: St. Clement, Origen, St. Iranaeus, St Augustine, St. Athenasius, St. Gregory, St. Cyril, Tertullian, St. Chrysostom, St. Cyprian, Theodoret. We call ourselves Catholic and Orthodox Anglicans. Our Metropolitan has said "Our doctrine is both Catholic and Orthodox." Our doctrine is so much that of the Greek Orthodox Church that I was asked to teach there.

Even earlier the Bishops of the C of E made a statement about doctrine.

II. From the books entitled "The Institution," and "A Necessary Doctrine, &c.," agreed on by the Bishops in England, A.D. 1537 and 1643: "All those things which were taught by the Apostles, and have been by a whole universal consent of the Church of Christ ever since that time taught continually, and taken always for true, ought to be received, accepted, and kept as a perfect doctrine Apostolic." III. From a Canon made by a Synod in England, A.D. 1571 Let Preachers above all things be careful that they never teach aught in a sermon to be religiously held and believed by the people, except that which is agreeable to the doctrine of the Old and New Testaments, and which has been collected from the same doctrine by the Catholic Fathers and ancient Bishops. [ed: which includes the Greek Fathers]

And here you see that the Anglican Church accepts the oral traditions of the Church as do the Orthodox.

IV. From a work of Archbishop Bramhall, entitled Schism Guarded: “We do not only admit oral traditions in general, as an excellent introduction to the doctrine of saving truth, and a singular help to expound the holy Scriptures, but also particular unwritten traditions, derived from the Apostles, and delivered unto us by the manifest testimony of the primitive Church, being agreeable to the holy Scriptures. The Apostles did speak by inspiration, as well as write; and their tradition, whether by word or writing, indifferently, was the Word of God, into which Faith was resolved. .... St. Augustine setteth us down a certain rule, how to know a true genuine Apostolical tradition: 'Whatsoever,' saith he, 'the universal Church doth hold, which has not been instituted by Councils, but always received, is most rightly believed to have been delivered by Apostolical authority.” Oxford ed. p. 382.

And from our Church web site: The word "Catholic" is often understood in opposition to the word "Protestant." However, this is both a recent and uniquely western perspective. In the ancient church, catholicism was understood to be the opposite of heresy, or false belief, and even today there are millions of Christians in Greece, Russia, and other parts of the world who consider themselves neither "Catholic" nor "Protestant," but "Orthodox."

From the Constitution of the ACC "The received Tradition of the Church and its teaching as set forth by “the ancient catholic bishops and doctors”, and especially as defined by the Seven Ecumenical Councils of the undivided Church, to the exclusion of all errors, ancient and modern.

http://en.wikipedia.org/wiki/Eucharist_(Catholic_Church)

Orthodox and Catholics believe that in the Eucharist the bread and wine are objectively transformed and become in a real sense the Body and Blood of Christ; and that after consecration they are no longer bread and wine: the consecrated elements retain the appearance and attributes of bread and wine but are in reality the body and blood of Christ.

Real Presence in regards to Catholic and Orthodox views is, essentially, transubstantiation.

The idea behind this is that the bread and wine actually become the body and blood of Christ after it has been consecrated.

This is quit different from the Lutheran view of "Real Presence".