1. The sacrificial character of the Holy Mass
1.1. In what sense is the Holy Mass a sacrifice?
The Holy Mass is a sacrifice in the proper sense, although “new" with respect to the sacrifices found in natural religions and in the ritual sacrifices of the Old Covenant. It is a sacrifice because the Holy Mass “re-presents" (makes present), in the “today" of the Church's liturgical celebration, the unique sacrifice of our redemption, and because it is its memorial and applies its fruit (cf. CCC , 1362 – 1367).
Each time the Church celebrates the Eucharist, she is called upon to accept the gift Christ offers her and thus to take part in her Lord's sacrifice, offering herself with Him to the Father for the salvation of the world. Hence we can say that the Holy Mass is the sacrifice of Christ and of the Church.
1.2 The Eucharist, sacramental presence of Christ's redemptive sacrifice
The Holy Mass is a true and proper sacrifice because of its direct relationship—its sacramental identity—with the unique, perfect and definitive sacrifice of the Cross.  This relationship was instituted by Christ at the Last Supper, when He gave the Apostles, under the species of bread and wine, his Body offered up in sacrifice and his Blood poured out for the remission of sins, anticipating in that memorial rite what took place on Golgotha a short while later. Since then the Church, under the guidance and with the power of the Holy Spirit, has never ceased fulfilling Christ's command to his disciples: Do this in memory of me (Lk 22:19; 1 Cor 11:24-25). For as often as you eat this bread and drink this cup, you proclaim the Lord's death until he comes (1 Cor 11:26; cf. Eph 5:2; Heb 9:26).
This sacramental proclamation of our Lord's Paschal Mystery is of special effectiveness. It is not only a sign or figure of Christ's redeeming sacrifice: it truly makes present his Person and the salvific event commemorated. “The Eucharist is the memorial of Christ's Passover, the making present and the sacramental offering of his unique sacrifice, in the liturgy of the Church which is his body" ( CCC , 1362).
Hence, when the Church celebrates the Eucharist, through the consecration of the bread and wine into the Body and Blood of Christ, the very Victim of Golgotha, now glorious, becomes present; the same Priest, Jesus Christ; the same act of sacrificial offering (the primordial offering of the Cross), inseparably united to the sacramental presence of Christ, an offering that is always present in the risen and glorious Christ.  Only the external manifestation of this self-giving changes. On Calvary, it was through Christ's passion and death on the Cross. In the Mass, it is through the memorial-sacrament: the double consecration of the bread and wine in the context of the Eucharistic Prayer, a sacramental image of the immolation on the Cross. 
In conclusion, the Last Supper, the sacrifice of Calvary, and the Eucharist are intimately related. The Last Supper was the sacramental anticipation of the sacrifice on the Cross. The Eucharist, which Christ instituted there, perpetuates (makes present) down through the centuries, wherever it is celebrated sacramentally, his unique redeeming sacrifice, so that all generations can enter into contact with Him and accept the salvation He offers to all humanity. 
1.3 The Eucharist, sacrifice of Christ and the Church
The Holy Mass is the sacrifice of Christ and the Church. “The Eucharist is also the sacrifice of the Church. The Church which is the Body of Christ participates in the offering of her Head. With him, she herself is offered whole and entire. She unites herself to his intercession with the Father for all men. In the Eucharist the sacrifice of Christ becomes also the sacrifice of the members of his Body. The lives of the faithful, their praise, sufferings, prayer, and work, are united with those of Christ and with his total offering, and so acquire a new value. Christ's sacrifice present on the altar makes it possible for all generations of Christians to be united with his offering" (CCC , 1368).
This teaching is affirmed by the living tradition of the Church, both in liturgical texts and in the Fathers and the Magisterium.  As the Second Vatican Council teaches: “Christ, indeed, always associates the Church with himself in this great work in which God is perfectly glorified and men are sanctified. The Church is his beloved bride who calls to her Lord, and through Him offers worship to the eternal Father."  The Church offers with Christ
The participation of the Church, the People of God, hierarchically structured, in the offering of the Eucharistic sacrifice is made possible by Christ's command, Do this in memory of me . This is reflected in the liturgical prayer, memores…offerimus…(tibi Pater) …gratias agentes…hoc sacrificium , frequently found in the Eucharistic prayers of the ancient Church,  and also present in the current Eucharistic prayers. 
As the words from the Eucharistic liturgy bear witness to, the faithful are not merely spectators at the act of worship offered by the celebrant. They all can and should join in the offering of the Eucharistic sacrifice, because by Baptism they have been incorporated into Christ and form part of a chosen race, a royal priesthood, a holy nation, God's own people (1 Pet 2:9). Christians form part of the new People of God in Christ, which he himself continuously gathers around himself, so that from one end of the earth to the other a perfect sacrifice may be offered to the Lord's name (cf. Mal 1:10-11). Christians offer not only the spiritual worship of the sacrifice of their own works and their entire existence, but also—in Christ and with Christ—the pure, holy, and immaculate Victim. Thus the common priesthood of the faithful is exercised in the Eucharist.
The faithful do not offer a different sacrifice from the sacrifice of Christ, for in uniting themselves with Him they make it possible for the Church's oblation to be incorporated with his, so that the Church's offering becomes the same offering as that of Christ. And it is He, Christ Jesus, who offers the spiritual sacrifice of the faithful, incorporating it into his own.
The Church is offered with Christ.
The Church, in union with Christ, not only offers the Eucharistic sacrifice, but is also offered up in Him, for as Christ's Body and Bride she is inseparably united to her Head and Spouse.
The teaching of the Fathers is very clear in this regard. For St. Cyprian, the “Church being offered" (the invisible oblation of the faithful) is symbolised in the liturgical offering of the gifts of bread and of wine mixed with a few drops of water, as the matter of the Sacrifice of the Altar.  For St. Augustine, in the Sacrifice of the Altar the whole Church is offered with her Lord, as the sacramental celebration itself makes clear: “This city fully redeemed, that is the assembly and society of the saints, is offered to God as a universal sacrifice by the High Priest who, in the form of a slave, offered himself for us in his passion to make us the body of so great a Head… Such is the sacrifice of Christians who 'though many, are one body in Christ' (Rom 12:5). The Church celebrates this mystery in the Sacrament of the Altar, as the faithful well know, where, in what she offers, she offers herself."  For St. Gregory the Great, the celebration of the Eucharist is a spur for us to imitate the example of our Lord, offering our lives to the Father as Christ did; and thus the salvation that stems from our Lord's Cross reaches us too. “When we celebrate this Eucharistic Sacrifice, we should offer ourselves to God with a contrite heart, for we who celebrate the mysteries of the Lord's passion must imitate what we do. And thus the Victim takes our place before God, if we make an offering of ourselves." 
The Eucharistic liturgy clearly expresses how the Church, under the influence of the Holy Spirit, takes part in Christ's sacrifice. Look with favor on your Church's offering and see the Victim whose death has reconciled us to yourself. Grant that we, who are nourished by his Body and Blood, may be filled with his Holy Spirit and become one body, one spirit in Christ. May he make us an everlasting gift to you…  Similarly, in Eucharistic Prayer IV: Look upon this sacrifice which you have given to your Church; and by your Holy Spirit, gather all who share this one bread and one cup into the one body of Christ, a living sacrifice of praise .
The faithful's participation consists above all in uniting themselves interiorly to Christ's sacrifice, made present on the altar through the ministry of the celebrant. However, it can never be said that that the faithful “concelebrate" with the priest,  since it is only he who acts in persona Christi Capitis , in the person of Christ the Head of the Church. But through the common priesthood received at Baptism, the faithful truly share in the sacrificial offering. This interior participation should be shown externally: receiving Holy Communion (if in the state of grace); in the responses and prayers that the faithful pray with the priest; in their posture and also sometimes participating in the Readings and the Prayers of the Faithful
Therefore all the faithful are called to take part in the Holy Mass, putting into practice the true priesthood they share in. That is, they should have the intention of offering to the Father their own life and the struggle to free it from sin, with Christ, the immaculate Victim, repaying with filial love and thanksgiving all that they have received from Him. Thus divine charity, the current of Trinitarian love at work in the celebration of the Eucharist, will transform their whole existence.
The faithful should strive to make the Holy Mass “the center and root of their interior life,"  thus giving expression to their “priestly soul." As St Josemaria urged us: “Keep struggling, so that the Holy Sacrifice of the Altar becomes the center and root of your interior life, and so your whole day will turn into an act of worship—an extension of the Mass you have attended and a preparation for the next. Your whole day will then be an act of worship that overflows in aspirations, visits to the Blessed Sacrament and the offering up of your professional work and your family life." 
Masses celebrated with none of the faithful present still have a public and social character. The effects of each Mass reach all places and times. Therefore it is very fitting that priests celebrate Mass every day, even when none of the faithful can be present. 
2. The ends and fruits of the Mass
The Holy Mass, in as much as it is the sacramental “making present" of Christ's sacrifice, has the same ends as the sacrifice of the Cross.  These ends are: that of “latria," that is, to praise and adore God the Father, through the Son, in the Holy Spirit; the Eucharistic aim, to give thanks to God for creation and the redemption; the propitiatory aim, to make amends for our sins; and the impetratory end, to beseech God for his gifts and graces. This is expressed in various prayers that form part of the liturgical celebration of the Eucharist, especially in the Gloria and Credo, in various parts of the Eucharistic Prayer (Preface, Sanctus, Epiclesis, Anamnesis, intercessions, final Doxology), in the Our Father, and in the prayers proper to each Mass: the Collect, Prayer over the offerings, Prayer after Communion.
The “fruits" of the Mass are the effects that the saving power of the Cross, made present in the Eucharistic sacrifice, produces in souls when freely accepted with faith, hope and love for the Redeemer. These fruits entail a growth in sanctifying grace and a closer conforming of one's life to Christ.
These fruits of holiness do not equally reach everyone who takes part in the Eucharistic sacrifice, but vary in accord with the degree to which each one participates in the liturgical celebration and with the measure of their faith and devotion. Hence sharing in different ways in the fruits of the Holy Mass are: the whole Church, the celebrant and those who, united with him, come together for the Eucharistic celebration, those who, without taking part in the Mass, unite themselves spiritually to the celebrant; and those for whom the Mass is offered, whether living or dead. 
When a priest receives an offering to apply the fruits of the Mass for a particular intention, he is gravely obliged to offer the Mass for that intention. 
Catechism of the Catholic Church , 1356-1372.
John Paul II, Enc. Ecclesia de Eucharistia , 17 April 2003, 11-29.
Benedict XVI Apost. Exhort. Sacramentum Caritatis , 22 February 2007, 6-15, 34-65.
Congregation for Divine Worship and Discipline of the Sacraments, Instruction, Redemptionis Sacramentum , 25 March 2004, 36-47, 48-79.
St Josemaria Escriva, Homily, “The Eucharist, Mystery of Faith and Love," in Christ Is Passing By , 83-94.
Joseph Cardinal Ratzinger, God Is Near Us: The Eucharist, the Heart of Life , Ignatius Press.
 “The sacrifice of Christ and the sacrifice of the Eucharist are one single sacrifice " ( CCC , 1367).
 “In the liturgy of the Church, it is principally his own Paschal Mystery that Christ signifies and makes present. During his earthly life, Jesus announces his Paschal Mystery by his teaching and anticipates it by his actions. When his Hour comes, he lives out the unique event in history which does not pass away: Jesus dies, is buried, rises from the dead and is seated at the right hand of the Father 'once for all.' His Paschal Mystery is a real event that occurred in our history, but it is unique: all other historical events happen once, and then they pass away, swallowed up in the past. The Paschal Mystery of Christ, by contrast, cannot remain only in the past, because by his death he destroyed death, and all that Christ is—all that he did and suffered for all men—participates in the divine eternity, and so transcends all times while being made present in them all. The event of the Cross and Resurrection abides and draws everything towards life" (CCC , 1085) .  The sacramental sign of the Eucharist does not cause again or reproduce the reality that is made present. It does not renew the bloody sacrifice of the Cross, because Christ has risen and death no longer has dominion over him (Rom 6:9). Nor does it cause in Christ anything that he does not already possess fully and definitively: it does not involve a new act of immolation and sacrificial offerings by Christ in glory. The Eucharist simply makes present a pre-existing reality: the Person of Christ, the Incarnate Word, who was crucified and has risen, and, in Him, the sacrificial act of our redemption. The sign gives Him a new way of being present, a sacramental presence, making possible the Church's participation in the Lord's sacrifice.
 “As often as the sacrifice of the Cross by which 'Christ our Pasch is sacrificed' (I Cor 5:7) is celebrated on the altar, the work of our redemption is carried out " (Const. Lumen Gentium , 3).
 Cf. CCC, 1368-1370.
 Vatican II, Const . Sacrosanctum Concilium , 7.
 Cf. Eucharistic prayer from the Apostolic Tradition of St. Hippolytus; Anaphora of Addai and Mari; Anaphora of St. Mark.  Cf. Roman Missal, Eucharistic Prayer I ( Unde et memores y Supra quae ); Eucharistic Prayer III ( Memores igitur; Respice, quaesumus e ipse nos tibi ). Similar expressions are found in Eucharistic Prayers II and IV.
 St Cyprian, Letter 63:13.
 St Augustine, City of God, 10:6.
 St Gregory the Great, Dialogue 4.61.1.
 Roman Missal, Eucharistic Prayer III , Respice, quaesumus e Ipse nos tibi .
 Pius XII, Enc. Mediator Dei , Dz 3850; Congregation for Divine Worship and Discipline of the Sacraments, Instruction Redemptionis Sacramentum , 42.
 St Josemaria Escriva, Christ is Passing By , 87.
 St Josemaria Escriva, The Forge , 69.
 Council of Trent, Teaching on the Holy Sacrifice of the Mass, ch. 6, Dz 1747; Vatican II, Decree Presbyterorun Ordini s, 13; John Paul II, Enc. Ecclesia de Eucharistica , 31; Benedict XVI , Sacramentum Caritatis , 80.
 This identity of ends is based not only on the intention of the celebrating Church, but above all on the sacramental presence of Jesus Christ himself, who continues to renew the ends for which he offered his life to the Father on Calvary (cf. Rom 8:34; Heb 7:25).
 This special prayer of intercession does not imply anything automatic about salvation: grace is not applied mechanically to these members of the faithful, but rather through their union with God by faith, hope and love.
 Cf. CCC , 945- 958. With this particular application of the Mass's fruits, the celebrant does not exclude the blessings of the Eucharistic sacrifice from reaching other members of the Church, and all humanity, but rather he simply includes some of the faithful in a special way.