After Paul has made a great case against us with regards to our unrighteousness before God (because of our sin) we get to the good news in Romans 3:21-26.
But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it (21)
Here is the good news. We were told in verse 20 that we could not justify ourselves (make ourselves righteous) by the Law. But God has provided us a way to obtain righteousness before him. This way was foretold by the Old Testament prophets and the Law pointed to it.
the righteousness of God through faith in Jesus Christ for all who believe.(22)
Here is the way. We can obtain righteousness before God through faith in Jesus Christ. We are reconciled with God through our faith. To believe in Jesus Christ means to trust in him, rely on him, obey him. Genuine faith is not a one-time statement of belief, but an ongoing condition.
For there is no distinction: for all have sinned and fall short of the glory of God, (22-23)
We are all sinners and have no hope of attaining righteousness before God on our own.
and are justified by his grace as a gift, through the redemption that is in Christ Jesus, (24)
Justified means to be declared righteous. It is an unmerited (by grace) gift from God through Jesus’ death on the cross. The payment for our penalty was made by Jesus – we are redeemed by His blood.
whom God put forward as a propitiation by his blood, to be received by faith. (25)
God’s offended holiness was appeased by the Jesus’ sacrifice. Our sins were placed on Jesus (who knew no sin) and his righteousness was imputed to us who believe. Jesus died in our place, he paid the penalty for our sins. God’s wrath is no longer on us if we accept his sacrifice through faith.
This was to show God's righteousness, because in his divine forbearance he had passed over former sins. (25)
God could have passed judgment on Adam and Eve immediately (and with that destroyed the human race), but he withheld His judgment until Christ to offer us His mercy.
It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus (26)
God’s great plan of salvation allowed Him to be just while also merciful to us since Jesus was punished in our place. Our sin was given to Jesus on the cross and his righteousness is given to us upon faith. Through faith in the Lord Jesus, we are saved and we are justified.
For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, Romans 1:16
I delight greatly in the LORD; my soul rejoices in my God. For he has clothed me with garments of salvation and arrayed me in a robe of righteousness, as a bridegroom adorns his head like a priest, and as a bride adorns herself with her jewels. Isaiah 61:10