Holy Communion augments our union with Christ. The principal fruit of receiving the Eucharist in Holy Communion is an intimate union with Christ Jesus. Indeed, the Lord said: "He who eats my flesh and drinks my blood abides in me, and I in him." (John 6:56)From there the Catechism goes on to describe graces bestowed in reception of the Eucharist such as renewal of baptismal grace (1392), spiritual nourishment (1394), separation from sin (1393-1395), and commitment to the poor (1397). These benefits are all proper to the recipient.
But the power of the Eucharist transcends the individual recipient. In fact the spiritual life of the world depends on the celebration of the Eucharist. How can this be so?
The extended power of the Eucharist in Scripture
One of the many passages in Scripture referring to the Eucharistic sacrifice comes from St. Paul:
1 Corinthians 10:16 The cup of blessing which we bless, is it not a participation in the blood of Christ? The bread which we break, is it not a participation in the body of Christ?So in the Eucharist, we unite with Christ's sacrifice as well as the Church--which Paul calls Christ's body (Col. 1:24; Eph. 1:22-23, et al).
Those who comprise the Church are extended throughout the world (e.g. 1 John 2:2, Rev. 7:9). The bread which we break (1 Cor. 10:16) brings eternal life (John 6:51) and also unites us to the Church (1 Cor. 10:16). Since we are united to that Church, when we receive the graces of Christ's sacrifice, the whole Church is united to the graces of that sacrifice. That includes anyone in the Church, anywhere, past, present, or future. We know that Christ's sacrifice transcends time in this way because His death was a historical event which saves both those in the New Testament and those in the Old (Mat. 27:52).
So when we participate in Christ by taking the "cup of blessing which we bless" and the "bread which we break" we are suffering with Him in a time-transcending extension of the once for all historical sacrifice on Calvary. And since Christ's sacrifice is the means to salvation for the Church, the entire Church in all ages and places benefits whenever we participate in that sacrifice.
Participation in Christ's suffering and unity of the body
Participation in Christ's suffering is very native to the New Testament. Paul said "we suffer with him in order that we may also be glorified with him" (Rom. 8:17). He told the Philippians how he longed to "share his sufferings." The Apostle Peter echoed the same theology:
1 Peter 4:13 But rejoice in so far as you share Christ's sufferings, that you may also rejoice and be glad when his glory is revealed.This concept of rejoicing in suffering with Christ contributes to our understanding of how the Eucharist benefits the entire body of Christ. Paul writes:
1 Corinthians 12:26 If one member suffers, all suffer together; if one member is honored, all rejoice together.And Paul further emphasized the indivisible unity of the "one body" of the Church (Eph 4:4, 1 Cor. 12:12). He ties this concept directly into the Eucharist one verse after 1 Corinthians 10:16 quoted above:
1 Corinthians 10:17 Because there is one bread, we who are many are one body, for we all partake of the one bread.
Further reinforcing the nature of the Eucharistic sacrifice as Christ's, Paul goes on to describe this taking of the bread and wine a participation "of the table of the Lord" (1 Cor. 10:21). He has already called the Eucharist a participation in the very blood of Christ's suffering. Referencing the "table of the Lord" fortifies his teaching that the Eucharist is indeed the sacrifice. He compares the "table" to the sacrificial "altar" of the Jews (1 Cor. 10:18). And he further compares it to "what pagans sacrifice" (1 Cor. 10:19). In the Old Testament, the concept of the Lord's "table" was the term used to describe where the Israelites offered sacrifice (Ezek. 44:15-16) which was the prefigurement of Christ's superior sacrifice perpetuated in the Eucharist (e.g. Heb. 9:23).
So adding all this up, what do we see? We see that participating in the Eucharist is a share in Christ's blood and the Church. His sacrifice is, of course, the source of salvation for the whole world. Therefore, since we sacramentally participate in the same suffering of His sacrifice by partaking in the Eucharist, the entire Church in every age and place benefits. After all, what happens to one member happens to all members since the body is united.
Thus, in partaking in the Eucharist we also show love of neighbor. When we rejoice, they rejoice. The devil's desire is the exact opposite of the Eucharist. The Eucharist is the grains of wheat united in "one bread, one body," bringing the Church everywhere together. The devil's desire is to "sift" the faithful "like wheat" that is not united (Luk. 22:31). The Eucharist brings unity to the Church. The devil brings division to the Church, and he incurs defeat whenever the Eucharist is celebrated.
Confirmed in Catholic teaching
The Catholic Church teaches how the entire Church throughout the world and in every age benefits whenever the Eucharistic sacrifice is celebrated. Here are some examples from official Catholic resources (emphasis mine):
For in [the Eucharist] Christ perpetuates in an unbloody manner the sacrifice offered on the cross, offering himself to the Father for the world's salvation through the ministry of priests–The Constitution on the Sacred Liturgy, Instruction on the Worship of the Eucharistic Mystery, 9.c.3. Quoted in Vatican Council II: The Conciliar and Post Conciliar Documents, edited by Austin Flannery, O.P., Liturgical Press, Collegeville, MN, 1977, p. 103Remember, Lord, those who have died and have gone before us marked with the sign of faith, especialy those for whom we now pray, N. et N. May these, and all who sleep in Christ, find in your presence light, happiness, and peace.–Eucharistic Prayer I, Roman Canon, Mass of the 1970 MissalLord, may this sacrifice, which has made our peace with you, advance the peace and salvation of all the world. Strengthen in faith and love your pilgrim Church on earth; your servant, Pope N., our Bishop N., and all the bishops,with the clergy and the entire people your Son has gained for you. Father, hear the prayers ofthe family you have gathered herebefore you. In mercy and love unite all your children wherever they may be. Welcome into your kingdom our departed brothers and sisters, and all who have left this world in your friendship.–Eucharistic Prayer III, Roman Missal, 3rd EditionThen, we pray [in the anaphora] for the holy fathers and bishops who have fallen asleep, and in general for all who have fallen asleep before us, in the belief that it is a great benefit to the souls on whose behalf the supplication is offered, while the holy and tremendous Victim is present. . . . By offering to God our supplications for those who have fallen asleep, if they have sinned, we . . . offer Christ sacrificed for the sins of all, and so render favorable, for them and for us, the God who loves man.–St. Cyril of Jerusalem, quoted in Catechism of the Catholic Church, #1371[T]he priest alone can complete the building up of the Body in the eucharistic sacrifice. Thus are fulfilled the words of God, spoken through His prophet: "From the rising of the sun until the going down thereof my name is great among the gentiles, and in every place a clean oblation is sacrificed and offered up in my name". In this way the Church both prays and labors in order that the entire world may become the People of God, the Body of the Lord and the Temple of the Holy Spirit, and that in Christ, the Head of all, all honor and glory may be rendered to the Creator and Father of the Universe.That the Sacrifice of the Mass is propitiatory both for the living and the dead.....Wherefore, not only for the sins, punishments, satisfactions, and other necessities of the faithful who are living, but also for those who are departed in Christ, and who are not as yet fully purified, is it rightly offered, agreebly to a tradition of the apostles.If any one saith, that the sacrifice of the mass is only a sacrifice of praise and of thanksgiving; or, that it is a bare commemoration of the sacrifice consummated on the cross, but not a propitiatory sacrifice; or, that it profits him only who receives; and that it ought not to be offered for the living and the dead for sins, pains, satisfactions, and other necessities; let him be anathema.The souls of the departed can, however, receive “solace and refreshment” through the Eucharist, prayer and almsgiving. The belief that love can reach into the afterlife, that reciprocal giving and receiving is possible, in which our affection for one another continues beyond the limits of death—this has been a fundamental conviction of Christianity throughout the ages and it remains a source of comfort today.–Pope Benedict XVI, Spe Salvi, 48
EDIT 10/23/12 TO ADD: The offering up of the Holy Mass benefits not only the saints for whom [in whose honor] it is said, but the whole Church of God in Heaven, on earth and in Purgatory.
–St. John Vianney, the Curé of Ars (quoted in Prayers and Heavenly Promises compiled from approved sources by Joan Carroll Cruz, p. 9)