One of the most unfortunate misunderstandings about the Catholic teaching on the Eucharist is that it consists of sacrificing Jesus Christ again - that at every Mass Christ is continually killed, continually suffering, continually dying, day in and day out. This misconception is partially due to the malice of certain detractors, and partially to an honest confusion over the relation of the Mass to the Sacrifice of the Cross. The Catholic Church does teach that the Eucharist is truly a sacrifice, and that it is truly our Lord Jesus Christ; therefore it is not too much of a stretch for those unfamiliar with Catholic teaching to wrongly assume that we believe our Lord is being killed at every Mass. While it is beyond the scope of this article to give a comprehensive treatment of the theology of the Eucharist, let us at least lay this bugbear to rest by showing that the Church does not and has never taught that Christ is sacrificed again in the Mass, as well as elucidate the true Catholic teaching on this particular point.
The Sacrifice of the Mass and Calvary
Catholic doctrine holds that Christ is truly present in the Sacrament of the Eucharist; in western tradition, His true presence begins from the time of the consecration and then endures so long as the sacramental species persist. Because it is Christ Himself who is present in the sacrament, the Mass does not simply consist of the worship, praise or offerings of the assembly, but rather of the very offering of Christ Himself to God the Father for the salvation of men. In the Mass, Christ Himself is present, and Christ Himself is offered. He offers Himself by virtue of His high priesthood; He is the offering by virtue of being the "Lamb of God, who takes away the sins of the world" (John 1:29). Thus, the Mass is truly an offering - the offering of Christ Himself.
But the only offering that Christ makes to God is of Himself, His whole being without reserve, which He offered for us on the cross. Christ, of course, died only one time upon the cross. This is a plain fact of history, as well as an affirmation of the Sacred Scriptures in multiple places:
"Christ also died once for our sins, the just for the unjust: that he might offer us to God, being put to death indeed in the flesh, but enlivened in the spirit" (1 Pet. 3:18).
"Knowing that Christ rising again from the dead, dies now no more, death shall no more have dominion over him. For in that he died to sin, he died once; but in that He lives, He lives unto God" (Rom. 6:9-10).
"So Christ, having been offered once to bear the sins of many, will appear a second time, not to deal with sin but to save those who are eagerly waiting for him" (Heb. 9:28).
It is a plain truth of the faith that Christ died His atoning death but once upon the cross.
It is this death upon the cross, this one event in history, that constitutes the formal act of Christ's sacrifice. In time, this act was carried out only once, as the above Scripture passages make clear. It is this sacrifice which makes our reconciliation with God possible. Though Jesus died only once, He is perpetually interceding for us before the throne of God, pleading His blood and the perfection of His offering to God on our behalf. "Christ Jesus, who died, yes, who was raised from the dead, who is at the right hand of God, who indeed intercedes for us" (Rom. 8:34).
This heavenly offering is the principle act of the Mass, where the one sacrifice of Christ is offered to God. This is a function of Christ's high priesthood. In the Epistle to the Hebrews, it is stated that because Christ is a perpetual high priest, He is ever offering Himself to God, and by virtue of this He is "able to save those who draw near" to Him. The author of Hebrews states:
"The former priests were many in number, because they were prevented by death from continuing in office; but he holds his priesthood permanently, because he continues for ever. Consequently he is able for all time to save those who draw near to God through him, since he always lives to make intercession for them" (Heb. 7:23-25).
Christ died on the cross only once; but because He ever lives, and because He is a perpetual high priest, He is always offering this sacrifice to God - and God is perpetually pleased by it because of the perfection of the offering, both in the priest and the victim. This is why, in the Resurrection, Christ still retains His wounds - these wounds are the signs of His sacrifice, which are forever presented to God. Thus, the offering of the cross and the offering of the Mass are both sacrificial, but the manner of the offering is different. The sacrifice of the cross was bloody, the perpetual offering of Christ to the Father in the Mass unbloody - nevertheless, both are sacrifices inasmuch as both flow from the one offering Christ makes of Himself to the Father.
We need to make an important distinction here: While the sacrifice of the Mass and the sacrifice of the cross are said to be the same sacrifice, this is meant in a very qualified sense. Well-meaning but poorly informed Catholics often get this point confused; knowing there is a link between the cross and the Mass, they errantly assume that the sacrifice of the cross is the sacrifice of the Mass in the historical sense, which in turn leads to all sorts of absurdities. For example, I keenly recall one woman many years ago explaining that at the Mass we actually "time travel" back to the cross, as if the offering of Christ in the Eucharist was His actual historical death. Her explanation of the Mass to her children was that they were traveling through time.
This misunderstanding was due to poor understanding of the relationship between the sacrifice of the cross and the sacrifice of the Mass. The two are the same sacrifice inasmuch as the priest is the same (Christ) and the victim is the same (Christ). Thus, the Mass and the Cross are the same sacrifice in the sense that someone who sees two showings of the same play in two successive nights has seem the same play, inasmuch as the cast and story remain the same. It does not mean, however, that in watching the play on night two one is somehow simultaneously witnessing the play from night one. Night one is night one, and night two is night two, even though they retain a very intimate connection.
Similarly, the Mass is grounded in the death of Christ on the cross - it is only because Christ died on the cross that He can eternally present that sacrifice to the Father - but the historical, bloody death of Christ on the cross is logically and theologically distinct from the unbloody presentation of that sacrifice. This is why the traditional formulations in the old catechisms teach that the Mass is the Sacrifice of the New Law in which Christ, through the ministry of the priest, offers Himself to God in an unbloody manner under the appearances of bread and wine. It is the manner of the offering which is distinct.
Is Christ offered at every Mass? Yes. Is this offering a true sacrifice? Yes. Is Christ slain at every Mass. No. He died once and dies no more.
The theology is admittedly intricate, especially for a Protestant or someone without any prior study of sacramental theology. It is sufficient to note that our above discussion should put to rest Protestant objections that we believe Christ is "killed" at every Mass. Still, in order to make the point more clearly, let us turn to history, especially the history of the Middle Ages and the Tridentine period, in order to make it clear that the Church has never taught that Christ dies at every Mass.
The idea that Christ is re-sacrificed in every Mass is said to be a perversion of the Gospel introduced during the Middle Ages, which to the opponents of the Church is the age of universal ignorance and superstition. To get a general idea of medieval thought on this issue, we can turn to Peter Lombard. Lombard (1100-1160) was the Bishop of Paris and most renowned theologian of the Middle Ages. Let us examine Peter Lombard's understanding of this question in his famous theological treatise, the Sentences. Remember that Peter Lombard was of such authority in the Middle Ages that all aspiring professors of theology were obliged to compose a commentary on Lombard's work in order to obtain their degree. Lombard can thus be taken as the authoritative exposition of medieval theology. In Book IV of the Sentences, he takes up the question of the nature of the Eucharistic sacrifice. Notice the emphasis on the continual offering of Christ, but with the distinction on the different modes or manners of offering:
The question is posed, whether what the priest does is properly called “sacrifice” or “offering” (immolatio), and whether Christ is offered (immoletur) daily, or was offered only once. The answer can be made briefly, that that which is offered and consecrated by the priest is called sacrifice and offering (oblatio), because it is the memory and representation of the true sacrifice and holy offering (immolatio) made on the altar of the Cross. And Christ died once on the Cross, and there was offered (immolatus) in Himself; but daily He is offered in the Sacrament, because in the Sacrament, there takes place the remembrance of what was done once...… what is done on the altar is and is called a sacrifice; and Christ was offered once, and is offered daily, but in one manner then, in another now. 
The Council of Trent, summoned to clarify the teaching of the Holy Catholic Church on the Mass against the errors of the 'Reformers', reaffirmed the distinctions noted by Peter Lombard. Note its citation of Hebrews 7:25, which we quoted above:
"And forasmuch as, in this divine sacrifice which is celebrated in the Mass, that same Christ is contained and immolated in an unbloody manner, who once offered Himself in a bloody manner on the altar of the cross; the holy Synod teaches, that this sacrifice is truly propitiatory and that by means thereof this is effected, that we obtain mercy, and find grace in seasonable aid, if we draw nigh unto God [cf. Heb. 7:25], contrite and penitent, with a sincere heart and upright faith, with fear and reverence. For the Lord, appeased by the oblation thereof, and granting the grace and gift of penitence, forgives even heinous crimes and sins. For the victim is one and the same, the same now offering by the ministry of priests, who then offered Himself on the cross, the manner alone of offering being different." 
This same teaching and distinctions are found in the writings of the Church Fathers as well. St. Augustine of Hippo teaches that the Mass is a true and proper sacrifice, for the very reason that it has a "real resemblance" to the historical death of Christ upon the Cross:
"Was not Christ once for all offered up in His own person as a sacrifice? And yet, is He not likewise offered up in the sacrament as a sacrifice, not only in the special solemnities of Easter, but also daily among our congregations; so that the man who, being questioned, answers that He is offered as a sacrifice in that ordinance, declares what is strictly true? For if sacraments had not some points of real resemblance to the things of which they are the sacraments, they would not be sacraments at all" .
In noting the "real resemblance" of the sacraments to that which they signify, he is referring to what has been since called the 'sign value' or 'signification' of the sacrament. In the Eucharist, the wine signifies the pouring out of Christ's blood and the broken bread His broken body upon the cross. This signification via the proper elements actually makes present that reality which is signified; thus, the offering of the Mass with the proper matter and form actually makes present Christ and His one offering, which is why the Mass is truly a sacrifice. This is basic sacramental theology - the Mass is truly the sacrifice of Christ, not because it is identical with the historical sacrifice of the cross or because Christ is dying again, but because of the "real resemblance" between the two manners of offering which bring forth the sacramental realities. It is noteworthy that where Augustine writes “He is sacrificed”, he uses the present infinitive “immolari”, expressing a continual action - the ongoing, perpetual unbloody offering of Christ to the Father.
There is an intimate link between the sacrifice of the cross and that of the Mass. Both are offerings of Christ the high priest; in both cases, Christ is also the victim. Because of this "real resemblance", it is right and proper to call the Mass a sacrifice in the true and proper sense, understanding that it is nothing else than the sacrifice of Christ to God the Father for the sins of the world. The sacrifice is the same, though the manner in which it is presented is different - one bloody, the other unbloody; one happening once in history, the other perpetually - which means that, while Christ is continually offering Himself to the Father in the sacrifice of the Mass, He is not slain daily. He "dies no more", as St. Paul says in the Epistle to the Romans. In fact, if He were not alive forever more, He could not make this perpetual intercession at the right hand of the Father.
Ad majorem Dei gloriam.
 Peter Lombard, Sentences, Lib. IV, Dist. 12
 Council of Trent, Session 22, Chapter 2
 St. Augustine of Hippo, Letter 98:9