This is my understanding of the Eucharist:
In the Eucharist, Christ turns bread and wine into His Body and Blood, thus he makes his whole self present, and - because he is the same yesterday, today and tomorrow - his death is present there (as is his ressurection, his whole self), and his death is shown forth - ‘proclaimed’, as St Paul said - and is present in the mystical separation of the body and blood. He only died once, so by making present his glorious and risen body and blood - mystically separated - the one same sacrifice of his body and blood is made present and so takes place on the altar, by the power of the Holy Spirit and through the Priesthood - he makes himself present as the Victim, and as Priest he (through the priest who is
in persona Christi, and by the Eucharistic Prayer) offers “his body and blood, the acceptable sacrifice which brings salvation to the whole world” (Eucharistic Prayer IV).
All so that we might receive the fruits of the sacrifice of the cross in time and space - here at this time, at this altar, and so Jesus can turn us into his body that we receive, both individually and as a church, [thus ‘the work of our redemption is carried out’-Lumen Gentium,3] so that we might become like him and so live no longer for ourselves but for him - by dying to sin (in our lives) by being united to him who has died for our sins, and living the life of grace with him who is risen and lives in us; so that we might have eternal life, starting now, within us, in the form of Sanctifying Grace.
Pope John Paul II gives an explanation of the Eucharist in Ecclesia de Eucharistia:
vatican.va/holy_father/special_features/encyclicals/documents/hf_jp-ii_enc_20030417_ecclesia_eucharistia_en.html
Here’s part of it (underlining added):
THE MYSTERY OF FAITH
11. “The Lord Jesus on the night he was betrayed” (1 Cor 11:23) instituted the Eucharistic Sacrifice of his body and his blood. The words of the Apostle Paul bring us back to the dramatic setting in which the Eucharist was born. The Eucharist is indelibly marked by the event of the Lord’s passion and death, of which it is not only a reminder but the sacramental re-presentation. It is the sacrifice of the Cross perpetuated down the ages.9 This truth is well expressed by the words with which the assembly in the Latin rite responds to the priest’s proclamation of the “Mystery of Faith”: “We announce your death, O Lord”.
The Church has received the Eucharist from Christ her Lord not as one gift – however precious – among so many others, but as the gift par excellence, for it is the gift of himself, of his person in his sacred humanity, as well as the gift of his saving work. Nor does it remain confined to the past, since “all that Christ is – all that he did and suffered for all men – participates in the divine eternity, and so transcends all times”.10
When the Church celebrates the Eucharist, the memorial of her Lord’s death and resurrection, this central event of salvation becomes really present and “the work of our redemption is carried out”.11 This sacrifice is so decisive for the salvation of the human race that Jesus Christ offered it and returned to the Father only after he had left us a means of sharing in it as if we had been present there. Each member of the faithful can thus take part in it and inexhaustibly gain its fruits. This is the faith from which generations of Christians down the ages have lived. The Church’s Magisterium has constantly reaffirmed this faith with joyful gratitude for its inestimable gift.12 I wish once more to recall this truth and to join you, my dear brothers and sisters, in adoration before this mystery: a great mystery, a mystery of mercy. What more could Jesus have done for us? Truly, in the Eucharist, he shows us a love which goes “to the end” (cf. Jn 13:1), a love which knows no measure.
Paul VI’s document Mysterium Fidei is also about the Eucharist:
vatican.va/holy_father/paul_vi/encyclicals/documents/hf_p-vi_enc_03091965_mysterium_en.html
Note the terminology sometimes used; it is not as often said now as it was formerly that the Sacrifice of Calvary is ‘repeated’ in the Eucharist, but this term was used in the same sense that ‘re-presented’ is now.
