1. After God has once received us into his family, it is not that he may regard us in the light of servants, but of sons, performing the part of a kind and anxious parent, and providing for our maintenance during the whole course of our lives. And, not contented with this, he has been pleased by a pledge to assure us of his continued liberality. To this end, he has given another sacrament to his Church by the hand of his only-begotten Son—viz. a spiritual feast, at which Christ testifies that he himself is living bread (John 6:51), on which our souls feed, for a true and blessed immortality… First, then, the signs are bread and wine, which represent the invisible food which we receive from the body and blood of Christ. For as God, regenerating us in baptism, ingrafts us into the fellowship of his Church, and makes us his by adoption, so we have said that he performs the office of a provident parent, in continually supplying the food by which he may sustain and preserve us in the life to which he has begotten us by his word. Moreover, Christ is the only food of our soul, and, therefore, our heavenly Father invites us to him, that, refreshed by communion with him, we may ever and anon gather new vigour until we reach the heavenly immortality. But as this mystery of the secret union of Christ with believers is incomprehensible by nature, he exhibits its figure and image in visible signs adapted to our capacity, nay, by giving, as it were, earnests and badges, he makes it as certain to us as if it were seen by the eye; the familiarity of the similitude giving it access to minds however dull, and showing that souls are fed by Christ just as the corporeal life is sustained by bread and wine. We now, therefore, understand the end which this mystical benediction has in view—viz. to assure us that the body of Christ was once sacrificed for us, so that we may now eat it, and, eating, feel within ourselves the efficacy of that one sacrifice,—that his blood was once shed for us so as to be our perpetual drink. This is the force of the promise which is added, “Take, eat; this is my body, which is broken for you” (Mt. 26:26, &c.). The body which was once offered for our salvation we are enjoined to take and eat, that, while we see ourselves made partakers of it, we may safely conclude that the virtue of that death will be efficacious in us. Hence he terms the cup the covenant in his blood. For the covenant which he once sanctioned by his blood he in a manner renews, or rather continues, in so far as regards the confirmation of our faith, as often as he stretches forth his sacred blood as drink to us.
10. The sum is, that the flesh and blood of Christ feed our souls just as bread and wine maintain and support our corporeal life. For there would be no aptitude in the sign, did not our souls find their nourishment in Christ. This could not be, did not Christ truly form one with us, and refresh us by the eating of his flesh, and the drinking of his blood. But though it seems an incredible thing that the flesh of Christ, while at such a distance from us in respect of place, should be food to us, let us remember how far the secret virtue of the Holy Spirit surpasses all our conceptions, and how foolish it is to wish to measure its immensity by our feeble capacity. Therefore, what our mind does not comprehend let faith conceive—viz. that the Spirit truly unites things separated by space. That sacred communion of flesh and blood by which Christ transfuses his life into us, just as if it penetrated our bones and marrow, he testifies and seals in the Supper, and that not by presenting a vain or empty sign, but by there exerting an efficacy of the Spirit by which he fulfils what he promises. And truly the thing there signified he exhibits and offers to all who sit down at that spiritual feast, although it is beneficially received by believers only who receive this great benefit with true faith and heartfelt gratitude. For this reason the apostle said, “The cup of blessing which we bless, is it not the communion of the blood of Christ? The bread which we break, is it not the communion of the body of Christ”? (1 Cor. 10:16.) There is no ground to object that the expression is figurative, and gives the sign the name of the thing signified. I admit, indeed, that the breaking of bread is a symbol, not the reality. But this being admitted, we duly infer from the exhibition of the symbol that the thing itself is exhibited. For unless we would charge God with deceit, we will never presume to say that he holds forth an empty symbol. Therefore, if by the breaking of bread the Lord truly represents the partaking of his body, there ought to be no doubt whatever that he truly exhibits and performs it. The rule which the pious ought always to observe is, whenever they see the symbols instituted by the Lord, to think and feel surely persuaded that the truth of the thing signified is also present. For why does the Lord put the symbol of his body into your hands, but just to assure you that you truly partake of him? If this is true let us feel as much assured that the visible sign is given us in seal of an invisible gift as that his body itself is given to us.
11. I hold then (as has always been received in the Church, and is still taught by those who feel aright), that the sacred mystery of the Supper consists of two things—the corporeal signs, which, presented to the eye, represent invisible things in a manner adapted to our weak capacity, and the spiritual truth, which is at once figured and exhibited by the signs. When attempting familiarly to explain its nature, I am accustomed to set down three things—the thing meant, the matter which depends on it, and the virtue or efficacy consequent upon both. The thing meant consists in the promises which are in a manner included in the sign. By the matter, or substance, I mean Christ, with his death and resurrection. By the effect, I understand redemption, justification, sanctification, eternal life, and all other benefits which Christ bestows upon us. Moreover, though all these things have respect to faith, I leave no room for the cavil, that when I say Christ is conceived by faith, I mean that he is only conceived by the intellect and imagination. He is offered by the promises, not that we may stop short at the sight or mere knowledge of him, but that we may enjoy true communion with him. And, indeed, I see not how any one can expect to have redemption and righteousness in the cross of Christ, and life in his death, without trusting first of all to true communion with Christ himself. Those blessings could not reach us, did not Christ previously make himself ours. I say then, that in the mystery of the Supper, by the symbols of bread and wine, Christ, his body and his blood, are truly exhibited to us, that in them he fulfilled all obedience, in order to procure righteousness for us— first that we might become one body with him; and, secondly, that being made partakers of his substance, we might feel the result of this fact in the participation of all his blessings.
John Calvin, Institutes of the Christian Religion, Book IV, Chapter 17, §1 & 10-11 (All the stuff in between is really good too and I would encourage you to read it.)