2nd Corinthians 5:21
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.
Simply put, to justify is to declare righteous, to make one right with God. Justification is God’s declaring those who receive Christ to be righteous, based on Christ’s righteousness being imputed to the accounts of those who receive Christ.
But now a righteousness from God, apart from law, has been made known, to which the Law and the prophets testify. This righteousness from God comes through faith in Jesus Christ to all who believe. There is no difference, for all have sinned and fall short of the glory of God, and are justified freely by his grace through the redemption that came by Christ Jesus. God presented him as a sacrifice of atonement, through faith in his blood. He did this to demonstrate his justice, because in his forbearance he had left the sins committed beforehand unpunished—he did it to demonstrate his justice at the present time, so as to be just and the one who justifies those who have faith in Jesus.
This sums up justification nicely, justification happens at the point of salvation. We do not become justified, we are pronounced justified when we are saved, right at that point. Some scholars argue that it happens right before.
Got Questions says,
Our righteousness comes from placing our faith in the finished work of Jesus Christ. His sacrifice covers our sin, allowing God to see us as perfect and unblemished. Because as believers we are in Christ, God sees Christ’s own righteousness when He looks at us. This meets God’s demands for perfection; thus, He declares us righteous—He justifies us.
Which is a great summary.