Substance theory is an ontological theory which argues that every object has a "substance" which is distinct from its "accidental" aspects. In other words, every object has both substantial properties and "accidental properties". "Substance" in this sense is a core concept of both ontology and metaphysics.
For example: a chair may be blue, tall, wide or narrow, made of wood or plastic. These properties are called "accidental" (in modern usage, we might be more inclined to say "incidental" or "circumstantial"), i.e. not part of the essential properties which make the object a chair. The essential properties of a chair might be specified, depending on your definition of "chair", as "sittability", "stability", and "movability".
This distinction, while valid in some ways, can lead to a false dichotomy if misapplied. For example, although the exact shape of a chair is not an "essential" property, if you take the pieces of wood which make up a chair and rearrange them, you could easily end up with something which is no longer a chair (except in a loose past-tense usage: "That stool over there is the chair I took apart last week.") "Chairness" is not an intrinsic property of the materials comprising the chair, but arises out of a combination of those intrinsic properties and how they are arranged. Even "intrinsic" physical properties arise from the manner in which the atoms and molecules of the construction materials are arranged.
The trap arises when one makes a decision about which properties are "essential" and which are incidental, and then uses that distinction as the sole basis for further conclusions: "Well, wood isn't an essential property of a chair, since some chairs are made of plastic or metal -- so I'm going to make a chair just like this one, but out of rubber, and it will still be a chair." It may look like a chair -- if you point at it and ask someone what it is, they will likely recognize the chair-ness of its shape -- but if you try to use it as one, you may well find that it collapses under you and is not really usable as a chair. Is it still a chair? Only if your definition of "chair" does not include being able to sit on it without making it collapse.
St. Thomas Aquinas used substance theory in articulating the theology of the Eucharist, particularly Transubstantiation: the "accidental properties" of the bread and wine do not change, but their essences change from bread and wine to "the Body and Blood of Christ".
In so doing, he clearly falls partway into the dichotomy trap shortly before taking a left turn down a rabbit-hole: he draws a firm distinction between the "accidental" (incidental) properties of the bread and wine, which he apparently defines as being all of their observable properties, and then claims that they have somehow acquired a substantial property of being something else besides (or perhaps in addition to) bread and wine.
In other words: he first claims that all the observable properties of wafers are somehow irrelevant to the wafer's "essence" and that changes to that "essence" would therefore in no way affect the observable properties of the wafer, and then he claims that the wafer-blessing somehow accomplishes such changes.
This is obviously nonsense; even if he somehow turned out to be correct, there is no way he (or anyone else) could determine if this were the case, since the changes are claimed to be unobservable. Furthermore, he has failed to define what the "essence" of the Body of Christ is or how it is different from the "essence" of the wafer. His statement is both baseless (how can Aquinas – or anyone – tell that this actually happens as predicted?) and meaningless (exactly how has the wafer changed?).
This sort of ontological handwaving is nonetheless a common characteristic of much theological discussion, and is given far too much credibility.
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- Catholic Encyclopedia: a very muddy definition