The Catholic doctrine of transubstantiation means that in the Eucharist, when the priest speaks Christ’s words of consecration (“this is my body”, “this is my blood”) over the bread and wine on the alter, the power of Christ is invoked, by the authority he gave to his apostles and they delegated to their successors and they delegated to the priests whom they consecrate, and it changes the bread and wine on the alter to become the body and blood of Christ. (This is sometimes called the “real presence.”) Much difficulty arises over exactly what is meant because the bread doesn’t turn into muscle tissue and the wine doesn’t develop red blood cells.
The Eastern Orthodox basically just say “it’s a mystery” and leave it at that. (I liked the styling I saw someplace, “eeeet’sss aaaaa myyyysssterrrryyyyy”.) The Catholic Church says that it’s a mystery, but it gives a few helpful details. You can actually see this in the word “transubstantiation.”
“Transubstantiation” is derived from two words: “trans” and “substance”. “Trans” meaning “change” and “substance” being that part of being which is not the accidents. Accidents, in this case, not meaning “something unintended” but rather the properties a thing has which, if they were changed or removed, would not make the thing something else. A chair might be made out of wood, but if you made it out of plastic it would still be a chair. The ability to hold up someone sitting is the substance of a chair, the material it is made out of is an accident (again, not in the colloquial sense of accident but in a technical sense). You can also do the reverse. You can take the wood a chair is made out of and rearrange it into a collection of splintery spikes protruding up. It has the same accidents (the wood), but the substance has changed. “Transubstantiation” just means that the accidents (the gluten, starch, etc. in the bread and the water, sugar, alcohol, etc. in the wine) remain the same but the substance—what it is—is what has changed.
Or, to put this more simply: in the Eucharist, the body and blood of Christ has the same chemical composition as bread and wine. Something to consider, when trying to understand this, is that a living human being has exactly the same chemical composition as a human corpse.