New Life in Jesus Christ: Whereas the sins of all people were forgiven on the Cross, receiving new and eternal life happens only when you trust Jesus Christ as Savior. All people have an inner spirit (also called your "heart" or your "nature"), and an outer mortal body. Two important things take place when you trust Jesus:  God completely removes your old sinful inner nature and replaces it with a brand new, clean heart and  Jesus comes to live in you and joins Himself eternally to you. You become a brand new person in your inner spirit, righteous and possessing eternal life because Jesus lives in you.
New Heart: the old is out and the new has come
Ezekiel 36:26 Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of stone from your flesh and give you a heart of flesh. (This is the great promise of the New Covenant as given by the Prophet Ezekiel.)
Romans 6:6 knowing this, that our old self was crucified with Him, that our body of sin (old sinful nature) might be done away with, that we should no longer be slaves to sin.
2 Corinthians 5:17 Therefore if anyone is in Christ, he is a new creature; old things passed away; behold new things have come.
New Life: Jesus comes to live in you making you righteous in your inner spirit and giving you eternal life.
Romans 5:17 For if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ.
1 John 5:11 And the testimony is this, that God has given us eternal life, and this life is in His Son.
1 John 5:12 He who has the Son has the life; he who does not have the Son of God does not have the life.
The Result: You are free from sin and completely righteous in your born again heart. You can never sin in your inner spirit because Jesus lives in you and He can never sin.
1 John 3:9 No one who is born of God practices sin, because His seed abides in him; and he cannot sin, because he is born of God.
2 Peter 1:4 For by these He has granted to us His precious and magnificent promises, in order that by them you might become partakers of the divine nature, having escaped the corruption that is in the world by lust.
Romans 6:7 for he who has died is freed from sin.
Romans 6:22 But now having been freed from sin and enslaved to God, you derive your benefit, resulting in sanctification, and the outcome, eternal life.
Your Life as a Believer: Your inner self is righteous but your outer mortal body has not been changed and can sin. This is the reason you live by faith. Faith is essentially letting Christ in you live His life out through you.
Romans 8:10 If Christ is in you, though the body is dead because of sin, yet the spirit is alive because of righteousness.
Romans 1:17 ... as it is written, “But the righteous man shall live by faith."
2 Corinthians 5:7 For we walk by faith, not by sight.
Hebrews 11:6 And without faith it is impossible to please him, for whoever would draw near to God must believe that he exists and that he rewards those who seek him.
John 15:5 "I am the vine, you are the branches; he who abides in Me and I in him, he bears much fruit, for apart from Me you can do nothing."
Galatians 2:20 I have been crucified with Christ; and it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by faith in the Son of God, who loved me and gave Himself up for me.