This question is part of a larger theological discussion among Christians in a branch of theology known as Soteriology.
Unfortunately there isn't a unanimous answer among all Christians on this question.
However we can broadly speak of two schools of thought:
Synergism - An individual's salvation involves a cooperation between the individual and God.
Monergism - An individual's salvation involves the work of God alone.
Rather than trying to further explain all the various answers from different denominations and branches of the Christian faith, I'll simply offer my view as a Christian from the Lutheran tradition.
Lutherans believe that salvation is monergistic, it is the work of God alone, apart from human effort, merit, or works. Indeed, on account of sin even the human will is broken, fallen, and bent away from God so that man is unable to take a step toward God, man cannot choose God, man cannot do anything to aid himself. Therefore man is entirely dependent upon God's own goodness, God's own kindness, mercy, grace, and love which is for us, freely ours, through Jesus Christ.
Therefore our salvation is accomplished, objectively speaking, in the Person and work of Jesus. By Christ's own perfect obedience He satisfies God's righteousness, and by His death and resurrection He defeats the powers of sin, death, hell, and the devil and actively and objectively justifies--sets right--sinful human beings before God. Therefore our sins are forgiven, we are reconciled to God, etc.
God Himself takes what Christ has accomplished for us once and for all and appropriates it, applies it to us, as individuals by the working of the Holy Spirit through the Means of God's Grace, Word and Sacrament. That is, the reason why Jesus commands that His Church preach the Gospel and administer the Sacraments is because this is how God has chosen to deliver to us, as individuals, the finished, accomplished, saving work of Jesus Christ.
When we hear the Gospel preached, God Himself acts and creates faith in us, through which we are justified, and turned toward God in His mercy. When we are baptized God Himself acts to unite us to Christ, by which we are buried with Him, die with Him, and are raised up together with Him to new life as new creations, born again, now alive to God in Christ and by the power of the Holy Spirit. Where we were formerly enemies toward God on account of our sin, we are now in Christ adopted children of God, having received a new life and identity in Jesus; so that what belongs to Christ is ours as a pure gift. His righteousness is now our righteousness as a gift; His Divine Sonship is now ours as grace as adopted sons and daughters of God His Father. We have received the Holy Spirit who is alive in us, and at work in us, holding us together in Christ, with one another, in the grace of God, with the hope and promise of life everlasting. So that all which God accomplished in Jesus is now ours. Christ rose from the dead as the first-fruits of the resurrection, and we likewise will be raised up from the dead at the conclusion of history.
Thus salvation is, we say, by grace alone, through faith alone, on Christ's account alone.
Grace alone, because this is all accomplished by God in His own generous kindness toward us.
Faith alone, because the gift of faith which God gives us is that through which we receive all which God has for us in Christ.
Christ alone, because all these things are once and for all, accomplished by Him by His life, death, and resurrection for our benefit.
Thus all who hear the Gospel, all who who have been baptized, all who trust in Jesus Christ are, indeed, saved; and this is God's word and promise to us, what He has accomplished for us apart from us. God has accomplished this all, for us, and because it is His work, it is perfect and trustworthy; we can depend upon the works of God, whereas human works are feeble and accomplish nothing. Likewise it is God's word and promise to us, and when He speaks to us it is trustworthy and true, for He who speaks is Faithful and True. So when He says to us, "You belong to Me" we can believe it, be confident in it, and know that He who began this work will finish this work. And so now and forever we belong to Jesus Christ. And since we belong to Christ, we belong to God, and God keeps us and holds us, securing us in Himself--that is our salvation.
-CryptoLutheran