I see a claim on Catholic blogs every now and then concerning the physical presence of the Body and Blood of Christ in the Eucharist. No doubt there is a physical presence in the Eucharist, but it is of the accidents of bread and wine. Christ's presence is substantial and sacramental (as noted in canon 1513 of the Council of Trent) leaving no substance of bread and wine. However, the accidents are not the accidents of Christ's physical body (which would naturally be repugnant). The confusion goes back to the time of the scholastics, roughly the 9th century. It was resurrected after the Protestant Reformers also began to question the settled doctrine and Sacred Tradition.
A monk by the name of Radbert Paschasius made the claim that the Eucharist was, according to the Catholic Encyclopedia, "converted into the real body of Christ, into the very body which was born of Mary and crucified." A monk of the same Abbey by the name of Ratramnus argued that there was no conversion of the bread but that the body of Christ was present in a spiritual way. The latter was forced to concede his view although whether he actually taught the mere symbolic presence of Christ in the Eucharist is up for debate.
Later on, Berengarius of Tours clearly held an understanding that could not be reconciled with the doctrine, which later came to be known as Transubstantiation. He also believed in a spiritual (as well as intellectual) presence of Christ. So the question was not whether Christ presence was merely symbolic—that is, whether the Eucharist is an empty symbol or a symbol plus something more. The question was whether His presence was in spirit or in some greater degree. Protestants are divided on the view of symbolic or spiritual presence. Catholics and Orthodox hold that this presence is essential.
The term "Transubstantiation" was first used by Hildebert of Tours and found its way into official usage at the Fourth Lateran Council. St. Thomas Aquinas developed a fuller theology on Transubstantiation. However, this is not to say that the underlying concepts did not exist prior to the time of the scholastics. In fact, the metaphysical basis of this theology comes from the Aristotelian notions of essence (substance) and accidents. However, even many of the early Church Fathers demonstrate a fairly developed theology on this matter, particularly in the Greek Fathers of the East and with St. Ambrose in the West.
What, then, is Transubstantiation? It is when, during the consecration of the Eucharist, the substance of the bread and wine change into the substance of the Body and Blood of Christ while the accidents of bread and wine remain. The confusing word in this mix is "substance," which in common parlance refers to a thing's material. However, in Aristotelian metaphysics, the substance is that which "stands underneath" (substantia)—what most of us would now refer to as the essence of a thing—that which makes a thing what it is.
We are used to accidents changing. Bread that has been left out and is moldy is still bread (until the chemical composition completely breaks down). The accidents of a human being are two legs, two arms, genitalia, a torso, a head, and so on. The essence of a human being remains even if the accidents are altered. A person who loses a limb or is left without sexual organs because of an injury is still a human being. What's more, we expect the accidents of human being to change over time. We can even change both substance and accidents of a thing through chemical and phsyical processes. Burn a piece of bread, and eventually neither its accidents or substance remain.
Transubstantiation is the opposite of the process described above. It is when the change occurs to the substance or essence of a thing rather than to its accidents. What starts as bread and wine, through consecration by the words of Christ and the power of the Holy Spirit, become something else. They become Christ's Body and Blood in essence—not spiritually but essentially.
To nonbelievers, this seems like nonsense on stilts. Granted, much of what Catholic and Orthodox Christians believe when taken in detail seems nonsensical. Of course, so does much of modern science, unless one understands the thinking that leads up to the conclusion. This fact is why atheists don't fair well in debates against Christians who have considered their faith through the eyes of reason. (And the same can be said for Christians who do not question and test what they believe against reasonable and consistent standards.) However, I can understand this skepticism, particularly to people who only accept a materialist view of the universe.
What has always puzzled me is how Christians consider it impossible for the God who created the universe from nothing and who revealed Himself through the Incarnation to transform the essence of bread and wine into His Body and Blood. It seems like a rather minor miracle* for someone who fed multitudes with 5 to 7 loaves, raised people from the dead, Himself rose from the dead, and altered the very course of history in the short span of 33 years through a ministry preaching love, mercy, and repentance. God, who created all things and holds all things in existence, and who transformed death and suffering into salvation, certainly has the power to change the substance of His own creation.
*Not to mention the fact that it flies in the face of the obvious words of scripture: "Unless you eat the flesh of the Son of Man and drink his blood, you have no life within you." Usually Jesus explained when He was being unclear, but for some reason, He didn't "explain" His words in John 6:53 and was willing to let His disciples walk away. With the persistent witness of the early Church Fathers on the matter, it's a shame that many Christians dismiss 1450 years of Sacred Tradition.