Catholics (and the Orthodox, the Anglican communion, and some Lutheran groups) believe that Jesus is physically present in the Eucharist. This is based on the passages in Matthew 26:26 - 28, Mark 14: 22 - 24, Luke 22: 19 - 20, I Corinthians 11: 23 - 27, and John 6: 48 - 58.
Transubstantiation is a way of attempting to explain the reality of the Divine Presence in bread and wine. It started off with Aristotle and his parsing of the world into philosophical constructs. One of the areas that he pondered was the nature of “being.” How is it that something is and what is its “true” nature. He defined the “true” nature of any object as its substance.
Years passed, Aristotle faded into memory while the writings of his teacher, Plato, held on through the writings of Philo and Plotinus and others, but Aristotle’s concepts of substance (that which something truly is) and accident (the way that something appears in the world), held on in various formulations. Around the eleventh century, various Christian theologians who were struggling with a way to explain the Divine Presence began to use the concept that at the Consecration of the mass, the bread and wine underwent a conversion of substance, and while they continued to look and taste like bread and wine, they were now really the body and blood of Jesus. The word that these theologians used to describe this action was transubstantiation. A few years later, contact with the Muslim world brought back copies of the works of Aristotle into Europe and Scholastic Philosophy/Theology was born, with the high point being the use of Aristotelian metaphysics by Thomas Aquinas to explain (or re-explain) most of Christian theology. At that point, Thomas was able to take the word transubstantiation and give it full explanation and meaning in the context of the entire Aristotelian logic. Instead of simply being one attempt to explain the phenomenon, it became a key point in a rigorous system of thought.
Interestingly, the (Platonically influenced) Franciscans initially attempted to have the works of Aquinas suppressed because so much of it was counter to their own ideas based on the squishy thoughts of Plato. (I freely confess that I find Plato and his ideals to be fairly silly.) However, the rigor and thoroughness that Thomas had brought to his work (probably aided by some attitudes of “look at this new stuff!”) eventually caused the Thomistic approach to win out.
A couple of hundred years later, Martin Luther, an Augustinian monk (while the Franciscans had reluctantly joined the Dominicans in switching to an Aristotelian view, the Augustinians had resolutely held out for the Platonic approach that had 1400 years of Christian commentary), reviewed the teachings and decided that transubstantiation had been a mistake. (This was not his single big break with the church, but it was a contributory argument.) His proposal was that Jesus was physically present in the Eucharist alongside the forms of bread and wine, and the name given to that holding was consubstantiation. In the ensuing brouhaha, the RCC, at Trent, proclaimed that transubstantiation was the only valid explanation for the Divine Presence, and various people have been glaring at each other over the terms ever since.
Of course, after Luther split, others did as well, and not all those who left the RCC continued to believe in the Divine Presence, so now we have some Catholics thinking that only Catholics believe in the Divine Presence, which is not true.
One obvious problem with the proclamation that only the word transubstantiation can describe the phenomenon, is that philosophy has moved on from the time of the Scholastics and Thomas Aquinas. A person who reviews the Aristotelian/Scholastic world view and comes to the conclusion that it does not actually describe the world has no reason to accept the word transubstantiation as descriptive, yet the church teaching enshrines the word in history.
My view has always been to say, that given a Scholastic approach, Transubstantiation is the appropriate word, whether I accept Scholastic Theology or not–and to the extent that I do not embrace Scholasticism, the word Transubstantiation is not really relevant to the discussion of the Divine Presence, in which I do believe.