what is justification?

Let's let the Westminster Shorter Catechism answer that. A great section for those with knowledge of the Federal Vision controversy:

Q. 70. What is justification?
A. Justification is an act of God's free grace unto sinners, in which he pardoneth all their sins, accepteth and accounteth their persons righteous in his sight; not for anything wrought in them, or done by them, but only for the perfect obedience and full satisfaction of Christ, by God imputed to them, and received by faith alone.

Q. 71. How is justification an act of God's free grace?
A. Although Christ, by his obedience and death, did make a proper, real, and full satisfaction to God's justice in the behalf of them that are justified; yet inasmuch as God accepteth the satisfaction from a surety, which he might have demanded of them, and did provide this surety, his own only Son, imputing his righteousness to them, and requiring nothing of them for their justification but faith, which also is his gift, their justification is to them of free grace.

Q. 72. What is justifying faith?
A. Justifying faith is a saving grace, wrought in the heart of a sinner by the Spirit and Word of God, whereby he, being convinced of his sin and misery, and of the disability in himself and all other creatures to recover him out of his lost condition, not only assenteth to the truth of the promise of the gospel, but receiveth and resteth upon Christ and his righteousness, therein held forth, for pardon of sin, and for the accepting and accounting of his person righteous in the sight of God for salvation.

Q. 73. How doth faith justify a sinner in the sight of God?
A. Faith justifies a sinner in the sight of God, not because of those other graces which do always accompany it, or of good works that are the fruits of it, nor as if the grace of faith, or any act thereof, were imputed to him for his justification; but only as it is an instrument by which he receiveth and applieth Christ and his righteousness.

Q. 77. Wherein do justification and sanctification differ?
A. Although sanctification be inseparably joined with justification, yet they differ, in that God in justification imputeth the righteousness of Christ; in sanctification his Spirit infuseth grace, and enableth to the exercise thereof; in the former, sin is pardoned; in the other, it is subdued: the one doth equally free all believers from the revenging wrath of God, and that perfectly in this life, that they never fall into condemnation; the other is neither equal in all, nor in this life perfect in any, but growing up to perfection.