My first encounter with Catholicism was at an international conference on religion and economics in Michigan. Because many attendees were Catholic, daily Mass was administered by the local parish priest, who was also the President of the hosting institution. I attended one morning, and found it an interesting though unfamiliar spectacle. Later that day, however, I had a chance to converse with a number of devout Catholics who explained to me some of the Catholic doctrines that were so foreign to my Protestant ears.
The single doctrine that struck me as simultaneously the most objectionable and the most defensible was the doctrine of Eucharistic Adoration. After the communal hosts have been consecrated by the priest, but before they are set out to be consumed at Mass, the elements are sometimes set aside in a chapel, where devout Catholic may go to worship before them.
From a Protestant perspective, this practice is tantamount to institutionalized idolatry. However, as my Catholic friend was describing it, he noted that, by the Catholic teaching of transubstantiation, the consecrated hosts are no longer merely bread and wine, but are in reality the body and blood of Christ. With such an understanding, worship or adoration of the elements can only be seen as worship or adoration of Christ.
The doctrine of transubstantiation is often misunderstood among both Protestants and lay Catholics, mostly because it relies on Aristotelian concepts foreign to modern ears. From his mentor Plato, Aristotle inherited notions of the world of forms -- the objective reality that undergirds the realm of subjective perception. Loosely speaking, when you sit on a chair, you are sitting on a corrupted image of a Perfect chair, on a physical (and therefore inconstant) manifestation of the Form of a chair. Aristotle's concept of forms was modified slightly from Plato, but he still argued that there is a real distinction between reality and the realm of the senses.
This distinction, in the Aristotelian system, is between substances and accidents. Substances are the thing itself, its essence or "quiddity" (a Latinate word roughly meaning its "what-ness"). Accidents are properties of the thing that we use to identify or describe it, but which aren't essential to its definition. For instance, this chair is made of wood. Is wood essential to the substance of a chair? No, because a chair might also be made of metal or plastic. Accidents belong to the realm of the senses, but substances are the thing itself.
Therefore, when Catholics speak of transubstantiation, they speak of the accidents (the outward appearance and physical reality) of the bread and wine remaining the same, while their substances are supplanted by the substance (inner reality, essence) of the body and blood of Christ. Under the Catholic hermeneutic, the words "this is my body" and "this is my blood," are not merely metaphorical but in fact definitional: Christ stated that the bread and wine were to be substantially identified with Himself.
This is the foundation for the Catholic dogma of the Real Presence, and the Catholic discipline of Eucharistic adoration.
One common Protestant objection to the Real Presence is the argument that, if it's necessary for Christ to be present in the host and to be broken each time Mass is celebrated, that would severely undermine the sufficiency of the atoning grace that arose from His sacrifice on the cross.
This objection is well taken, but it seems to hinge on a rather crucial misunderstanding. As created beings, we live our whole lives in the contexts of time and space, and find it quite impossible to imagine anything outside them. But time and space are properties of the created order, and thus God as Creator transcends such limitations. In the original celebration of the Lord's Supper, Jesus referred to His body broken, and His blood poured out, prior to His crucifixion, yet He asked His disciples to eat and drink in remembrance of that future act. No wonder the disciples were confused!
God exists in what we might consider a timeless present: this is the meaning of His Name, the Tetragrammaton "I am that I am." In every moment He experiences and observes everything. This is, incidentally, the solution to the problem of divine foreknowledge and predestination. For now, it also resolves the issue of competing sacrifices. It seems to me that God is sacrificially present in the act of communion, not in addition to His sacrifice on Calvary, but in the same divine moment.
We partake of that single sacrifice, and share the same Holy Meal, without regard to our differences in space and time from the rest of the Church, because God is present throughout space and time.
This is, incidentally, a necessary counterpoint to the doctrine of the Real Presence. After all, as Christians we must concede that God is present everywhere and such presence is indeed real. But there is a difference in the act of communion. There is a visceral sacredness, an almost oppressive sense that this is truly a Holy Meal, just as surely as the ground beneath Moses' feet was holy ground. The transcendent Lord of the Universe is peeking out, as though from from behind the veil, and even that partial and opaque glimpse can still unravel the strings of our soul.