In the document the council begins by addressing the controversial matter of real presence. Is the real body and blood of the Lord Jesus Christ present in the bread and the wine of the Eucharist? The Catholic position is an unqualified "Yes." Christ is "truly, really, and substantially contained under the species of those sensible things" — that is, under the bread and wine that still look, feel, smell, and taste like bread and wine. This presence is not at all contradicted by the fact that Christ remains bodily at the right hand of the Father (still a common argument made against Transubstantiation).
The basis for this is the divine attribute of omnipresence, which the council suggests that Jesus shares even in his incarnate, resurrected and glorified body.
What's puzzling is that this is usually considered not a Tridentine view but a Lutheran one, and one of the major efforts of Counter-Reformation thinkers (like Bellarmine), was to attack it. This Lutheran view was that Christ was really present in the eucharist by virtue of the fact that His humanity participated in the omnipresence of his divinity. It's called the ubiquity of Christ. The idea is that there is a 'communication of properties' in Christ in three ways:
(1) idiomatic: where we ascribe the properties of the nature to the person;
(2) apotelesmatic: where we ascribe the properites of the person to the nature;
(3) auchematic: where we ascribe the properties of one nature to the other.
The real presence of Christ by ubiquity would be an example of (3).
This view was strongly opposed by Calvinists and Catholics alike, who, whatever their differences, share a similar understanding of the Chalcedonian definition, and regard this doctrine of ubiquity as a clear violation of it. Further, it has been pointed out by this reasoning, Christ would be bodily present in every cabbage in your garden, and the uniqueness of the institution is lost. (There are versions of the Lutheran doctrine in which this is not the case; for instance, 'volipresential' views hold that Christ's humanity in virtue of its union with the divine has the power to be present as He wills -- a virtual rather than actual omnipresence.)
The Tridentine view is rather different:
For neither are these things mutually repugnant,-that our Saviour Himself always sitteth at the right hand of the Father in heaven, according to the natural mode of existing, and that, nevertheless, He be, in many other places, sacramentally present to us in his own substance, by a manner of existing, which, though we can scarcely express it in words, yet can we, by the understanding illuminated by faith, conceive, and we ought most firmly to believe, to be possible unto God.
That is, Christ is present not naturally (by virtue of his humanity) but sacramentally (by virtue of the power of God). This sacramental presence is a key part of what it is to be assimilated to the Body of Christ, which is sacramental union with Christ, whereby through His presence we partake of His life.
The Calvinist opposition to ubiquity was along broadly similar lines. Like the Catholics they insisted that
(1) at the heart of the institution of the eucharist is an inexhaustible mystery;
(2) the key to this mystery is our union with Christ.
Unlike the Catholics, however, the Calvinists held that the presence of Christ was better understood in terms of reconciliation than in terms of bodily presence. That is, the Catholic view is that Christ has a unique mode of presence, by which he is substantially present in the eucharist, uniting believers to Himself. The Calvinists held that Christ is not substantially present in the eucharist; rather, His presence is the very act of uniting believers to Himself. It's usually called spiritual presence.
This Calvinist viewpoint eventually began to lose ground in Reformed churches to a Zwinglian viewpoint, in which there is no real presence at all, only a symbolic one: the Lord's Supper symbolizes Christ's body and blood, and nothing more. This has been an ongoing struggle in Reformed circles, with the Zwinglians usually having the upper hand (an upper hand, however, that they seem to have been steadily losing in past decades).
