The meaning of creation is determined by its supernatural end, which is union with God. However, sin profoundly upset the order of creation: mankind ceased to see the world as a work full of goodness and turned it into something equivocal. People placed their hope in creatures and established false earthly goals for themselves.
The purpose of Jesus Christ's coming into the world is to re-establish God's plan and lead the world to its true destiny of union with God. To do this, Jesus, the true Head of the human race,  took upon himself the whole of human nature degraded by sin, made this nature his own, and offered it as a Son to the Father. In this way Jesus restored to every human relationship and situation their true meaning, which is their dependence on God the Father.
This meaning or purpose of Christ's coming is fulfilled through the whole of his life, through each of the mysteries in his life, in which Jesus fully glorifies the Father. Each event and stage in Christ's life has a specific purpose ordered to this salvific goal. 
1.2. Meaning of the Mystery of the Cross
The real purpose of the mystery of the Cross is to cancel out the sin of the world (cf. Jn 1:29), which is absolutely necessary if we are to achieve filial union with God. This union is, as stated above, the ultimate goal of God's plan (cf. Rom 8:28-30).
Jesus rids the world of sin by taking it upon his own shoulders and destroying sin in the justice of his holy heart.  The mystery of the Cross consists essentially in the following.
a) He took our sins upon himself . This is seen, in the first place, in his passion and death as related in the Gospels. Since these events happened to the Son of God incarnate and not just to a man, however holy, they have a universal value and effectiveness that applies to the whole human race. In the Gospels we see that Jesus was given by the Father into the hands of sinners (cf. Mt 26:45) and that he himself allowed their wickedness to determine his fate. As Isaiah says in his powerful portrait of the “Suffering Servant":  He was oppressed and afflicted, yet he opened not his mouth; like a lamb that is led to the slaughter, and as a sheep before its shearers is dumb, so he opened not his mouth ( Is 53:7).
A lamb without blemish, Christ freely accepted the physical and mental suffering imposed on him by the injustice of sinners, and through it, he took upon himself all the sins of men, every offence committed against God. Every human affront is, in some way, the cause of Christ's death. In this sense we say that Jesus “bore" our sins on Golgotha (cf. 1 Pet 2:24).
b) He eliminated sin through his self-giving. Christ did not limit himself to bearing our sins, he also “destroyed" them. They were eliminated because he accepted his sufferings with filial righteousness, in obedient and loving submission to his Father God; and with innocent righteousness, as one who loves sinners even though they do not deserve it; seeking to forgive our offences out of love (cf. Lk 22:42; 23:34). He offered his sufferings and death to his Father on our behalf and for our forgiveness: With his stripes we are healed ( Is 53:5).
The fruit of the Cross is, therefore, the elimination of sin. We can make this fruit our own through the sacraments (especially sacramental Confession), and we will do so definitively after this life, if we have been faithful to God. The Cross offers all men and women the possibility of avoiding sin and of integrating Christ's sufferings and death into their own path to holiness.
2. The Cross reveals God's mercy and justice in Jesus Christ
God chose to save the world by way of the Cross, but not because he loves pain or suffering, since God only loves good and does good. He did not want the Cross with an unconditional will, like his will, for example, that creatures should exist, but he willed it praeviso peccato, presupposing sin. The Cross is there because sin exists. But love also exists. The Cross is the fruit of God's love in response to men's sins.
God chose to send his Son into the world to bring about the salvation of mankind through the sacrifice of his own life, and this tells us much about God himself. Specifically the Cross reveals to us the mercy and justice of God:
a) God's mercy . Holy Scripture frequently refers to the Father giving his Son into the hands of sinners (cf. Mt 26:54), not sparing his own Son. Through the unity of the divine Persons of the Trinity, the Father who sent him is always present in Jesus Christ, the incarnate Word. Therefore, behind Jesus' free decision to give his life for us, there is the Father's surrender of his beloved Son for us, handing him over to sinners; this surrender shows, more than any other gesture in the history of salvation, the Father's love for mankind and his mercy.
b) The Cross also reveals to us God's justice. This does not consist so much in making human beings pay for their sins as in setting us again on the path of truth and goodness, and restoring the gifts destroyed by sin. Christ's faithfulness, obedience and love towards his Father God; his generosity, charity and forgiveness of mankind, his brothers and sisters; his truthfulness, justice and innocence, maintained and reaffirmed at the moment of his passion and death, do all of this. They empty sin of its power to send us to hell, and open our hearts to holiness and justice, since he gives himself for us. God frees us from our sins through justice, Christ's justice.
As the result of Christ's sacrifice and through the presence of his saving power, we are always able to behave as children of God, whatever the situation in which we find ourselves.
3. Sinners caused the Cross to happen
Jesus knew from the beginning, in a way appropriate to the progress of his mission and of his human awareness, that his life was leading to the Cross. And he accepted it fully: he came to do the Father's will down to the very last detail (cf. Jn 19:28-30), and doing so led him to give his life as a ransom for many (Mk 10:45).
In carrying out the task his Father had entrusted to him, he met with the opposition of the religious authorities in Israel, who considered Jesus to be an impostor. “Some of the leaders of Israel accused Jesus of acting against the law, the temple in Jerusalem, and in particular against faith in the one God because he proclaimed himself to be the Son of God. For this reason they handed him over to Pilate so that he might condemn him to death" (Compendium , 113).
The people who condemned Jesus to death sinned by rejecting the Truth that is Christ. In reality, every sin is a rejection of Jesus and of the truth that he brought us from God. In that sense, every sin has its place in the Passion of Jesus. “The passion and death of Jesus cannot be imputed indiscriminately either to all the Jews that were living at that time or to their descendants. Every single sinner, that is, every human being is really the cause and the instrument of the sufferings of the Redeemer; and the greater blame in this respect falls on those above all who are Christians and who the more often fall into sin or delight in their vices" (Compendium , 117).
4. Sacrifice and Redemption
Jesus died for our sins (cf. Rom 4:25), to free us from them and redeem us from the slavery that sin introduced into mankind's life. Holy Scripture says that the passion and death of Christ are: a) a covenant sacrifice; b) a sacrifice of expiation, c) a sacrifice of atonement and reparation for sins, d) an act of the redemption and liberation of mankind.
a) Jesus, by offering his life to God on the Cross, instituted the New Covenant, that is to say, the new form of union of God with men that had been prophesied by Isaiah (cf. Is 42:6), Jeremiah (cf. Jer 31:31-33) and Ezekiel (cf Ez 37:26). The new Alliance is the covenant sealed in the body of Christ offered up for us, and in his blood shed for us (cf. Mt 26:27-28).
b) Christ's sacrifice on the Cross has an expiatory value, that is to say, the value of cleansing and purifying us from sin (cf. Rom 3:25; Heb 1:3; 1 Jn 2:2; 4:10).
c) The Cross is a sacrifice of atonement and reparation for sin (cf . Eph 2:16, 5:2; Phil 2:8-9; Heb 5:1-10, 13:11-12). Christ rendered to his Father the love and obedience that we human beings had denied him through our sins. His self-giving manifested justice and satisfied the fatherly love of God which we had rejected from the very beginning of history.
d) Christ's Cross is an act of the redemption and liberation of mankind. Jesus paid for our freedom with the price of his blood, that is, of his suffering and death (cf. 1 Pet 1:18). By giving his life he merited our salvation so as to incorporate us into the kingdom of heaven: He has delivered us from the dominion of darkness and transferred us to the kingdom of his beloved Son, in whom we have redemption, the forgiveness of sins ( Col 1:13-14).
5. Effects of the Cross
The main effect of the Cross is to eliminate sin and everything opposed to our union with God.
As well as destroying sin, the Cross also frees us from the devil, who, while remaining hidden, engineers the whole trauma of sin and eternal death. The devil can do nothing against those who are united to Christ (cf . Rom 8:31-39), and death ceases to be an eternal separation from God, and becomes instead the gateway to our final destiny (cf. 1 Cor 15:55-56).
The Cross removes all the obstacles and opens up the way of salvation and the possibility of grace for all mankind.
Together with Christ's Resurrection and his glorious Exaltation, the Cross is the cause of man's justification, that is, not only the destruction of sin and all the other obstacles, but also the infusion of new life (Christ's grace which sanctifies the soul). Each sacrament is a different way of participating in Christ's Pasch and of making our own the salvation that flows from it. Baptism in particular frees us from the death introduced by original sin and enables us to live the new life of the risen Christ.
Jesus is the one, universal cause of human salvation, the only mediator between God and men. Each saving grace given to men proceeds from Christ's life, and in particular, from his paschal mystery.
6. Co-redeeming with Christ
As stated above, the Redemption worked by Christ on the Cross is universal: it extends to the whole human race. But the fruit and merits of Christ's Passion need to be applied to each person, principally by means of faith and the sacraments.
Our Lord Jesus Christ is the one mediator between God and men (cf. 1 Tim 2:5). But God the Father has willed that we should not only be redeemed but also be co-redeemers (cf. Catechism , 618). He calls us to take up his Cross and follow him (cf. Mt 16:24), because he suffered for you, leaving you an example, that you should follow in his steps " (cf. 1 Pet 2:21).
Saint Paul writes:
a) I have been crucified with Christ; it is no longer I who live, but Christ who lives in me " ( Gal 2:20). To reach identification with Christ we have to embrace the Cross.
b) In my flesh I complete what is lacking in Christ's afflictions for the sake of his body, that is the church ( Col 1:24). We can be co-redeemers with Christ.
God has not chosen to free us from all the hardships in this life. By accepting them we can identify ourselves with Christ, merit eternal life and co-operate in the work of bringing the fruits of Redemption to others. Sickness and pain, offered to God in union with Christ, achieve great redeeming value, as does corporal mortification practised in the same spirit in which Christ suffered, freely and voluntarily, in his Passion: out of love, to redeem us, expiating for our sins. On the Cross Jesus Christ sets us an example of all the virtues:
a) charity: Greater love has no man than this, that a man lay down his life for his friends ( Jn 15:13);
b) obedience: He became obedient unto death, even death on a cross ( Phil 2:8);
c) humility, meekness and patience. He bore his sufferings without avoiding them or diminishing them, like a meek lamb (cf. Jer 11:19);
d) detachment from the things of earth: the King of kings and Lord of those who rule appears on the Cross naked, mocked, spat upon, scourged, crowned with thorns, for Love.
Our Lord chose to associate his Mother with the mystery of redemptive suffering more closely than anyone else (cf. Lk 2:35; Catechism , 618). Our Lady teaches us to stand by her Son's Cross. 
Catechism of the Catholic Church , 599-618
Compendium of the Catechism of the Catholic Church , 112-124
John Paul II, “The Redemptive Value of Christ's Passion," Catechesis: 7 September 1988, 8 September 1988, 5 October 1988, 19 October 1988, 26 October 1988.
John Paul II, “Christ's Death: its Redemptive Character," Catechesis: 14 December 1988, 11 January 1989.
St Josemaría, Homily “Christ's Death is the Christian's Life," in Christ is Passing By , 95-101.
 He is our Head because he is the Son of God and because he made himself one with us in everything except sin (cf. Heb 4:15)
 Christ's infancy, his life of work, his baptism in the Jordan, his preaching, and everything else, contributes to the Redemption of mankind. Referring to Christ's life in the town of Nazareth, St Josemaría said, “His hidden years are not without significance, nor were they simply a preparation for the years which were to come after – those of his public life. Since 1928 I have understood clearly that God wants our Lord's whole life to be an example for Christians. I saw this with special reference to his hidden life, the years he spent working side by side with ordinary men. Our Lord wants many people to ratify their vocation during years of quiet, unspectacular living" ( Christ is Passing By, 19).
 Cf. Col 1:19-22; 2:13-15; Rom 8:1-4; Eph 2:14-18; Heb 9:26
 The four poems dedicated to the mysterious “Servant of God" are a moving prophecy in the Old Testament of Christ's Passion. ( Is 42:1-9; 49:1-9; 50:4-9; 52:13 – 53:12)
 Cf. Josemaría Escrivá, The Way, 508.