What Is Transubstantiation?
following is primarily from
“The Catechism of the Council of Trent”
“The Catechism of the Council of Trent”
Meaning of the Real Presence
Jesus Christ Whole And Entire Is Present In The Eucharist
Here the pastor should explain that in this Sacrament are contained not only the true body of. Christ and all the constituents of a true body, such as bones and sinews, but also Christ whole and entire. He should point out that the word Christ designates the God-man, that is to say, one Person in Whom are united the Divine and human natures; that the Holy Eucharist, therefore, contains both, and whatever is included in the idea of both, the Divinity and humanity whole and entire, consisting of the soul, all the parts of the body and the blood, - all of which must be believed to be in this Sacrament. In Heaven the whole humanity is united to the Divinity in one hypostasis, or Person; hence it would be impious, to suppose that the body of Christ, which is contained in the Sacrament, is separated from His Divinity.
In Virtue Of The Sacrament
And In Virtue Of Concomitance
Pastors, however, should not fail to observe that in this Sacrament not all these things are contained after the same manner, or by the same power. Some things, we say, are present in virtue of the [Double] Consecration; for as the words of Consecration effect what they signify, sacred writers usually say that whatever the Form expresses, is contained in the Sacrament by virtue of the Sacrament. Hence, could we suppose any one thing to be entirely separated from the rest, the Sacrament, they teach, would be found to contain solely what the Form expresses and nothing more.
On the other hand, some things are contained in the Sacrament because they are united to those which are expressed in the Form. For instance, the words This is My Body, which comprise the Form used to Consecrate the bread, signify the body of the Lord, and hence the body itself of Christ the Lord is contained in the Eucharist by virtue of the Sacrament. Since, however, to Christ's body are united His blood, His soul, and His Divinity, all of these also must be found to coexist in the Sacrament; not, however, by virtue of the Consecration, but by virtue of the union that subsists between them and His body. All these are said to be in the Eucharist by virtue of concomitance. Hence it is clear that Christ, whole and entire, is contained in the Sacrament; for when two things are actually united, where one is, the other must also be.
Christ Whole And Entire Present Under Each Species
Hence it also follows that Christ is so contained, whole and entire, under either species, that, as under the species of bread are contained not only the body, but also the blood and Christ entire; so in like manner, under the species of wine are truly contained not only the blood, but also the body and Christ entire.
But although these are matters on which the Faithful cannot entertain a doubt, it was nevertheless wisely ordained [i.e. instituted by Christ Himself] that two distinct Consecrations should take place. First, because they represent in a more lively manner the Passion of our Lord, in - which His blood was separated from His body; and hence in the Form of Consecration we commemorate the shedding of His blood. Secondly, since the Sacrament is to be used by us as the food and nourishment of our souls, it was most appropriate that it should be instituted as food and drink, two things which obviously constitute the complete sustenance of the (human) body.
Christ Whole And Entire Is Present In Every Part Of Each Species
Nor should it be forgotten that Christ, whole and entire, is contained not only under either species, but also in each particle of either species. Each, says St. Augustine, receives Christ the Lord, and He is entire in each portion. He is not diminished by being given to many, but gives Himself whole and entire to each. 
This is also an obvious inference from the narrative of the Evangelists. It is not to be supposed that our Lord Consecrated the bread used at the Last Supper in separate parts, applying the Form particularly to each, but that all the bread then used for the Sacred Mysteries was Consecrated at the same time and with the same Form, and in a quantity sufficient for all the Apostles. That the Consecration of the [wine in the] chalice was performed in this manner, is clear from these words of the Saviour: Take and divide it among you.  [a]
What has hitherto been said is intended to enable pastors to show that the true body and blood of Christ are contained in the Sacrament of the Eucharist.
The Mystery of Transubstantiation
The next point to be explained is that the substance of the bread and wine does not continue to exist in the Sacrament after [the Double] Consecration. This truth, although well calculated to excite our profound admiration, is yet a necessary consequence from what has been already established.
Proof From The Dogma Of The Real Presence
If, after [the Double] Consecration, the true body of Christ is present under the species of bread and wine, since it was not there before, it must have become present either by change of place, or by creation, or by the change of some other thing into it. It cannot be rendered present by change of place, because it would then cease to be in Heaven; for whatever is moved must necessarily cease to occupy the place from which it is moved. Still less can we suppose the body of Christ to be rendered present by creation; nay, the very idea is inconceivable. In order that the body of our Lord be present in the Sacrament, it remains, therefore, that it be rendered present by the change of the bread into it. Wherefore it is necessary that none of the substance of the bread remain.
Proof From The Councils
Hence our predecessors in the faith, the Fathers of the General Councils of Lateran and of Florence, confirmed by solemn decrees the truth of this dogma. In the Council of Trent it was still more fully defined in these words: If any one shall say that in the most Holy Sacrament of the Eucharist the substance of the bread and wine remains, together with the body and blood of our Lord Jesus Christ, let hint be anathema. 
Proof From Scripture
The doctrine thus defined is a natural inference from the words of Scripture. When instituting this Sacrament, our Lord Himself said: This is My Body.  The word this expresses the entire substance of the thing present; and therefore if the substance of the bread remained, our Lord could not have truly said: This is My Body.
In St. John, Christ the Lord also says: The bread that I will give is my flesh, for the life of the world.  The bread which He promises to give, He here declares to be His flesh. A little after He adds: Unless you eat the flesh of the son of man, and drink his blood, you shall not have life in you.  And again: My flesh is meat indeed, and my blood is drink indeed.  Since, therefore, in terms so clear and so explicit, He calls His flesh bread and meat indeed, and His blood drink indeed, He gives us sufficiently to understand that none of the substance of the bread and wine remains in the Sacrament.
Proof From The Fathers
Whoever turns over the pages of the holy Fathers will easily perceive that on this doctrine (of transubstantiation) they have been at all times unanimous. St. Ambrose says: You say, perhaps, “this bread is no other than what is used for common food”. True, before consecration it is bread; but no sooner are the words of consecration pronounced than from bread it becomes the flesh of Christ. [a] To prove this position more clearly, he elucidates it by a variety of comparisons and examples.
In another place, when explaining these words of the Psalmist, Whatsoever the Lord pleased he hath done in heaven and on earth.  St. Ambrose says: Although the species of bread and wine are visible, yet we must believe that after consecration, the body and blood of Christ are alone there. 
Explaining the same doctrine almost in the same words, St. Hilary says that although externally it appear bread and wine, yet in reality it is the body and blood of the Lord. 
Why The Eucharist Is Called Bread After Consecration
Here pastors should observe that we should not at all be surprised, if, even after Consecration, the Eucharist is sometimes called bread. It is so called, first because it retains the appearance of bread, and secondly because it keeps the natural quality of bread, which is to support and nourish the body.
Moreover, such phraseology is in perfect accordance with the usage of the Holy Scriptures, which call things by what they appear to be, as may be seen from the words of Genesis which say that Abraham saw three men,  when in reality he saw three Angels. In like manner the two Angels who appeared to the Apostles after the Ascension of Christ the Lord into heaven, are called not Angels, but men. 
The Meaning of Transubstantiation
To explain this mystery is extremely difficult. The pastor, however, should endeavour to instruct those who are more advanced in the knowledge of Divine things on the manner of this admirable change. As for those who are yet weak in faith, they might possibly be overwhelmed by its greatness.
Transubstantiation - A Total Conversion
This conversion, then, is so effected that the whole substance of the bread is changed by the power of God into the whole substance of the body of Christ, and the whole substance of the wine into the whole substance of His blood, and this, without any change in our Lord Himself. He is neither begotten, nor changed, not increased, but remains entire in His substance.
This sublime mystery St. Ambrose thus declares: You see how efficacious are the words of Christ. If the word of the Lord Jesus is so powerful as to summon into existence that which did not exist, namely the world, how much more powerful is His word to change into something else that which already has existence? [a]
Many other ancient and most authoritative Fathers have written to the same effect. We faithfully confess, says St. Augustine, that before consecration it is bread and wine, the product of nature; but after consecration it is the body and blood of Christ, consecrated by the blessing.  The body, says Damascene, is truly united to the Divinity, that body which was derived from the virgin; not that the body thus derived descends from heaven, but that the bread and wine are changed into the body and blood of Christ. 
This admirable change, as the Council of Trent teaches, the Holy Catholic Church most appropriately expresses by the word Transubstantiation.  Since natural changes are rightly called transformations, because they involve a change of form; so likewise our predecessors in the faith wisely and appropriately introduced the term Transubstantiation, in order to signify that in the Sacrament of the Eucharist the whole substance of one thing passes into the whole substance of another.
According to the admonition so frequently repeated by the holy Fathers, the Faithful are to be admonished against curious searching into the manner in which this change is effected. It defies the powers of conception; nor can we find any example of it in natural transmutations, or even in the very work of creation. That such a change takes place must be recognised by Faith; how it takes place we must not curiously inquire.
No less of caution should be observed by pastors in explaining the mysterious manner in which the body of our Lord is contained whole and entire under the least particle of the bread. Indeed, discussions of this kind should scarcely ever be entered upon. Should Christian charity, however, require a departure from this rule, the pastor should remember first of all to prepare and fortify his hearers by reminding them that no word shall be impossible with God.  [a]
A Consequence Of Transubstantiation
The pastor should next teach that our Lord is not in the Sacrament as in a place. Place regards things only inasmuch as they have magnitude. Now we do not say that Christ is in the Sacrament inasmuch as He is great or small, terms which belong to quantity, but inasmuch as He is a substance [i.e. a quiddity; an ontological essence]. The substance [ontological essence] of the bread is changed into the substance [ontological essence] of Christ, not into magnitude or quantity; and substance, it will be acknowledged by all, is contained in a small as well as in a large space. The substance of air, for instance, and its entire nature must be present under a small as well as a large quantity, and likewise the entire nature of water must be present no less in a glass than in a river. Since, then, the body of our Lord succeeds to the substance of the bread, we must confess it to be in the Sacrament after the same manner as the substance of the bread was before Consecration; whether the substance of the bread was present in greater or less quantity is a matter of entire indifference [i.e. it is irrelevant to the point under discussion].
The Mystery of the Accidents without a Subject
We now come to the third great and wondrous effect of this Sacrament, namely, the existence of the species of bread and wine without a subject.
Proof From The Preceding Dogmas
What has been said in explanation of the two preceding points must facilitate for pastors the exposition of this truth. For, since we have already proved that the body and blood of our Lord are really and truly contained in the Sacrament, to the entire exclusion of the substance of the bread and wine, and since the accidents of bread and wine cannot inhere in the body and blood of Christ, it remains that, contrary to physical laws, they must subsist of themselves, inhering in no subject.
Proof From The Teaching Of The Church
This has been at all times the uniform doctrine of the Catholic Church; and it can be easily established by the same authorities which, as we have already proved, make it plain that the substance of the bread and wine ceases to exist in the Eucharist.
Advantages Of This Mystery
Nothing more becomes the piety of the Faithful than, omitting all curious questionings, to revere and adore the majesty of this august Sacrament, and to recognise the wisdom of God in commanding that these Holy Mysteries should be administered under the species of bread and wine. For since it is most revolting to human nature to eat human flesh or drink human blood, therefore God in His infinite wisdom has established the administration of the body and blood of Christ under the forms of bread and wine, which are the ordinary and agreeable food of man.
There are two further advantages: first, it prevents the calumnious reproaches of the unbeliever, from which the eating of our Lord under His visible form [i.e. appearance] could not easily be defended; secondly, the receiving Him under a form [i.e. appearance] in which He is impervious to the senses avails much for increasing our Faith. For Faith, as the well known saying of St. Gregory declares, has no merit in those things which fall under the proof of reason. 
The doctrines treated above should be explained with great caution, according to the capacity of the hearers and the necessities of the times. [a] (“The Catechism of the Council of Trent for Parish Priests”, Issued by Order of Pope Saint Pius V, Antonio-Michele Ghislieri [Friday, January 7, 1566 - Monday, May 1, 1572]. Translated by John Ambrose McHugh, O.P., S.T.M., Litt.D., and Charles Jerome Callan, O.P., S.T.M., Litt.D. Imprimatur: ? Patrick Cardinal Joseph Hayes, D.D., [b. at New York, New York on Wednesday, November 20, 1867 - d. at New York, New York on Sunday, September 4, 1938;, Archbishop of New York, Monday, March 10, 1919 - Sunday, September 4, 1938]. Part II, “The Sacrament of the Eucharist”, pp. 233-241.)
Sources Quoted above in “The Catechism of the Council of Trent for Parish Priests”:
[51: Quoted in John Gratian [12th Century], “Concordantia Discordantium Canonum” (Harmonizing Discordant Canons), a.k.a. “Decretum Gratiani” (Decretals of Gratian), Distinction 2, Chapter 74; Ambrosian Mass, Preface for the Fifth Sunday after Epiphany.]
[a: On the manner of the Real Presence see Saint Thomas Aquinas, O.P., [b. 1225 A.D. in Rocca Secca, Naples, Italy - d. Wednesday, March 7, 1274 A.D. in Fossa Nuova, Italy], Doctor of the Church: “Angelic Doctor”, “Common Doctor”; “Summa Theologica”, Part III, Question 74.]
[59: Quoted in John Gratian, “Decretum Gratiani” (Decretals of Gratian), Distinction 2, 74; cf. De Myster. 9; Paschasius Radbertus (Saint Paschal Radbert) [ b. at Soissons in 786 A.D. - d. in the Monastery of Corbie, c. 860 A.D.], “Liber De Sacramento Corporis et Sanguinis Christi” (“A Book on the Sacrament of the Body and Blood of Christ”), Book 1, Chapter 1, n. 2.]
[60: Saint Hilary of Poitiers, Bishop, Confessor, and Doctor of the Church [b. at Pictavium, Gaul (modern-day Poitiers, France, c. 310 A.D. - d. at Poitiers, France in c. 367 A.D.], De Trinitate”, lib. 8 (“On the Trinity”, book 8); quoted in John Gratian, “Decretum Gratiani” (Decretals of Gratian), Distinction 2, Chapters 70 and 82.]
[a: These words are taken from the work of Patriarch Saint Aurelius Ambrosius (Saint Ambrose) [b. at Augusta Treverorum, Gallia Belgica, Roman Empire - modern day - Trier, Germany in c. 340 A.D. - d. at Mediolanum, Italia annonaria, Roman Empire on Friday, April 4, 397 A.D.], Patriarch of Milan [Saturday, December 7, 374 - Friday, April 4, 397 A.D.], “De Sacramentis”, Book 4, Chapter 4 and Book 5, Chapter.4.]
[64: Saint John Damascene, a.k.a. Saint John of Damascus, Doctor of the Church [b. at Damascus, Bilad al-Sham, Umayyad Caliphate in c. 675 or 676 - d. at Mar Saba, Jerusalem, Bilad al-Sham, Umayyad Caliphate on Sunday, December 4, 749], “De Fide Orthodoxa” (“On the Orthodox Faith”), was the first work of Scholasticism, Book 4, Chapter 13. Concerning the Holyand Immaculate Mysteries of the Lord.]
[a: On Transubstantiation see Saint Thomas Aquinas, O.P., [b. 1225 A.D. in Rocca Secca, Naples, Italy - d. Wednesday, March 7, 1274 A.D. in Fossa Nuova, Italy], Doctor of the Church: “Angelic Doctor”, “Common Doctor”; “Summa Theologica”, Part III, Question 75, Articles 1-8.]
[a: On the accidents remaining in the Eucharist, see Saint Thomas Aquinas, O.P., “Summa Theologica”, Part III, Question 75, Article 5. Also see Saint Thomas Aquinas, O.P.,“Summa Theologica”, Part III, Question 77.]
For more information on this subject, We recommend the “Summa Theologica” of Saint Thomas Aquinas, especially Part III, Questions: 73. Of the Sacrament of the Eucharist; 74. Of the Matter of this Sacrament; 75. Of the Change of Bread and Wine into the Body and Blood of Christ; 76. Of the Way in Which Christ Is in this Sacrament; and 83. Of the Rite of this Sacrament.
“...the whole substance of the bread is changed into the whole substance of Christ's body, and the whole substance of the wine into the whole substance of Christ's blood. Hence this is not a formal, but a substantial conversion; nor is it a kind of natural movement: but, with a name of its own, it can be called transubstantiation.” Saint Thomas Aquinas, O.P., [b. 1225 A.D. in Rocca Secca, Naples, Italy - d. Wednesday, March 7, 1274 A.D. in Fossa Nuova, Italy], Doctor of the Church: “Angelic Doctor”, “Common Doctor”; “Summa Theologica”, Part III, Question 75, Articles 4.)
For the record, historically, the term “Transubstantiation” was used for the first time by Hildebert of Tours in about the year 1097 A.D.
Roman Catholics have no choice in this matter because all Roman Catholics are required to believe, under the censure of anathema (i.e. automatic excommunication) what the Council of Trent teaches:
“Canon 1. If anyone denies that in the Sacrament of the Most Holy Eucharist are contained truly, really and substantially the Body and Blood together with the Soul and Divinity of our Lord Jesus Christ and consequently the whole Christ, but says that He is in it only as in a sign, or figure or force; let him be anathema.” (Session 13, Thursday, October 11, 1551; emphasis added).
2. If anyone says that in the Sacred and Holy Sacrament of the Eucharist
the substance of the bread and wine remains conjointly with the Body and
Blood of our Lord Jesus Christ and denies that wonderful and singular change
of the whole substance of the bread into the Body and the whole substance
of the wine into the Blood, the appearances only of bread and wine remaining,
which change the Catholic Church most aptly calls “Transubstantiation”;
let him be anathema.” (Session 13, Thursday,
October 11, 1551; emphasis added).
“ANATHEMA then appears as the more solemn form of pronouncing or declaring excommunication.” (Rev. P. Charles Augustine Bachofen, O.S.B., D.D., Professor of Canon Law, “A Commentary on the New Code of Canon Law”, B. Herder Book Co., 1922, Volume VIII, Book V, Penal Code [Canons 2195 - 2414], Part II, Penalties, Section II, Penalties in Particular, Title VIII, Corrective Penalties or Censures, Chapter II, Censures in Particular, Penalties, Article I, Excommunication, Canons 2255-2256, p. 170.)
ANATHEMA: A thing or person struck by God’s malediction and intended for ruin. Cf. I Cor. 12:13; Rom. 9:3; Gal. l:8-9. Anathema, in actual Church discipline, is the term used for IPSO FACTO [i.e. automatic] excommunication incurred by those denying a solemnly defined Truth, as is concluded principally from the Dogmatic Canons of the Roman Catholic Council of Trent and the Vatican Council, (i.e. the Roman Catholic Council Vatican I). (Parente, Piolanti, Garofalo, Dictionary of Dogmatic Theology, “Anathema”.)