Then I saw a great white throne and Him who sat upon it, from whose presence earth and heaven fled away, and no place was found for them. And I saw the dead, the great and the small, standing before the throne, and books were opened; and another book was opened, which is the book of life; and the dead were judged from the things which were written in the books, according to their deeds. And the sea gave up the dead which were in it, and death and Hades gave up the dead which were in them; and they were judged, every one of them according to their deeds. (Revelation 20:11-13)
First note that Christians are not the dead, they are alive in Christ spiritually and physically, so they are not the ones being judged here. We see in verse 6 that all Christians are resurrected a clear thousand years before this white throne judgement. "Blessed and holy are those who share in the first resurrection. The second death has no power over them, but they will be priests of God and of Christ and will reign with him for a thousand years" (Revelation 20:6).
Jesus takes away the sins of the world (John 1:29). God has removed the sins of His people as far as the east is from the west (Psalm 103:12). Those who reject Christ don't have their sins taken away in this infinite fashion. God makes relationship with Himself available by not counting sins against anyone (2 Corinthians 5:19). Jesus takes away the sins of the world in this life in order to offer salvation freely without sin being in the way of anyone trusting Him.
Hebrews 9:26 says, Jesus came in the flesh to put away sin by the sacrifice of Himself. When someone trusts Christ their sin is put away forever. It is eradicated without even a possible future resurgence (see 1 John 3:9). If someone dies without trusting Christ their sin is taken back out and they are judged by their deeds.
In the next life God says, "Ok you didn't want My Son's sacrifice for your sins so now I am returning your sins to you and judging you by your deeds." God no longer remembers the sins of His people (Hebrews 8:12,10:17). Those who reject Him are not His people. To be just to those who reject His gift of salvation God judges them according to their deeds. It doesn't say He judges them according to the old covenant law. His judgement goes beyond the law so proper justice is given. "For the Lord searches all hearts, and understands every intent of the thoughts" (1 Chronicles 28:9).
For more about what happens on judgment day see:
Eternal Torment, Annihilation, Ultimate Reconciliation? How about a little of each?
Second Chances & Why God Killed