Transubstantiation is the
Catholic doctrine that during the
sacrament of
communion, the
communion wafer actually and quite literally becomes
the body of Christ in all ways but appearance. This is considered to be an honest-to-god
miracle, true
divine intervention in the
universe.
Of course, the skeptic will point out that if it changes in all ways but appearance, what's the big miracle? The Catholic responds that God has changed the substance of the wafer, that what it is has changed (though this can only be perceived through faith). But what does this mean, that the substance of the wafer has changed? In our modern empirical world view, we tend to go on what properties we see in an object to tell us what it is- if those properties change, the thing becomes another thing; if those properties stay the same, the thing remains what it is.
When early Christianity was forming into a cohesive unit with what might be called an academic wing, it needed some sort of formal beliefs about the nature of the physical world and took on Aristotle as a guide. Many Aristotelian notions were taken as nearly (and sometimes actually) dogmatic. One of these was Aristotle's metaphysics. Without going into great detail, Aristotle regards any given thing as being, well, a particular thing and furthermore having certain qualities. That is, the red door might have the property of being red, but it is a door- that is the one and only category that it fits in to.
So what happens according to the doctrine of transubstantiation is that God causes the communion wafer to keep all its properties while at the same time changing the substance of it, something that is naturally impossible according to Aristotle, thus the miracle.