What does "real presence" and transubstantiation mean today in the Catholic church?
--Real questions in Rocky River
There are two documents in the Catholic church, Constitution on the Sacred Liturgy  and General Instruction on the Roman Missal [1969/70] in paragraph 7 of each document, that say that Christ is present in four ways at Mass: in the assembly of the baptized; in the person of the ordained minister; in the proclamation of the scriptures; and in the consecrated bread and wine. Christ is present par excellence in the eucharistic elements.
Also, reflecting on the term "real presence," Pope Paul VI wrote in his encyclical letter Mysterium fidei (1965):
This presence is called real presence not to exclude the other kinds as though they were not real, but because it is real par excellence, since it is substantial, in the sense that Christ, whole and entire, God and man, becomes present.
It is helpful to remember from eucharistic theology that the word "substantial" is a Thomistic, neo-Aristotelian term; it does not mean "material" as it is often understood and used in "common" speech. Today we tend to think that substance is matter. During the scholastic age, the term was quite literally to mean "that which stands beneath" the appearance of the thing ("the accident").
The traditional Catholic understanding has always been that Christ is present in the eucharist, whole and entire, under the sign of bread and wine. The bread and wine do not cease to smell, taste, look, feel like bread and wine. Yet they become the body and blood of Christ. When Catholics worship the eucharist after the liturgy, they are worshiping Christ, still present under the forms of bread and wine not an idol.
The Council of Trent at its thirteenth session said in Decree on the Most Holy Eucharist (1551), Chapter IV, in The Christian Faith in the Doctrinal Documents of the Catholic Church:
... by the consecration of the bread and the wine there takes place a change of the whole substance of bread into the substance of the body of Christ our Lord and of the whole substance of wine into the substance of His blood. This change the holy Catholic Church has fittingly and properly named transubstantiation.