Rightousness

God made him who had no sin to be sin for us, so that in him we might become the righteousness of God.
(2. Corinthians 5:21)

To be righteous means to have a right standing before God. It means that when you stand before God, you don’t have to feel embarrassed because you have not been able to live up to His standards.
    One of the big difference between the Old and the New covenant is how righteousness is obtained.
    Under the old covenant it was something you worked for by living right and offering sacrifices according to the law. The problem was that nobody could was able to do this as perfectly as a perfect God required.

So Paul states in the epistle to the Romans:

As it is written: “There is no one righteous, not even one...
(Romans 3:10)

Because of sin we have a power within that is pulling us in the opposite direction of righteousness. So trying to live in a right standing before God was living in a constant battle against self.
    When people sinned, they could offer sacrifices for sin, but the sin within was not removed, so it was just a matter of time before they again needed to offer new sacrifices.

Another problem was that since sin within was not removed, man was not really righteous in the first place. This meant that even though God was gracious and forgave the wrong they had done, they could never really become righteous because they had already fallen short.

The new covenant
But things changed under the new covenant, but contrary to what many believe it was not God who changed.

Many have an image of God in the Old Testament as grumpy but now under the new covenant He has become happy. This is not the case, because according to James 1:17 God never changes. God has always been good,

“The Lord, the Lord, the compassionate and gracious God, slow to anger, abounding in love and faithfulness, maintaining love to thousands, and forgiving wickedness, rebellion and sin…
(Exodus 34:6-7)

So God has always been compassionate, gracious, slow to anger, loving, faithful and forgiving. What changed was the covenant, meaning the agreement between God and man. Under the old covenant righteousness was something you had to work for, but under the new covenant righteousness is something that is given to you; before you have done anything to attain it.

But now apart from the law the righteousness of God has been made known, to which the Law and the Prophets testify. This righteousness is given through faith in Jesus Christ to all who believe. There is no difference between Jew and Gentile,
Romans 3:21-22

So under the new covenant righteousness is something you can be given, even if you have not lived a righteous life, if you believe in Jesus. Here it’s not talking about a belief in the existence of Jesus, because even the devil has this kind of faith. It’s talking about a faith the becomes evident in your actions. This is why the Bible often portrait disobedience as the opposite of faith, and obedience as a sign of love, faith and belief.

Whoever believes in the Son has eternal life; whoever does not obey the Son shall not see life, but the wrath of God remains on him.
John 3:36

If you love me, you will keep my commandments… “If anyone loves me, he will keep my word… Whoever does not love me does not keep my words
John 14:15, 23-24

If we are to obtain righteousness by achieving it, we have already lost. For until the time in our life where we decide to start being good we have already done something wrong; therefor we are already unrighteous. And even if we for the rest of our life try to be really good, we have in our bodies the nature of sin that pulls us toward sin.

They only way we can obtain righteousness, is if God chooses to give it to us; and this is what He does in Christ Jesus.

Justified
To be justified mean that you declared not guilty in the charges someone has accused you of. The devil is accusing mankind before, but when we get justified, we are declared not guilty in these charges.

all have sinned and fall short of the glory of God, and all are justified freely by his grace through the redemption that came 
by Christ Jesus.
Romans 3:23-24

Paul says that the accusations against us was nailed to the cross.

And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross.
Colossians 2:13-14

How could God do this and still himself act rightly? After all He himself would not be righteous if He looked between the fingers when it came to our sin.

But the grace of God is not that He now chooses to take lightly on our sin. The grace of God is that He now solves our problem with sin. He does this through the new birth. The old guilty me was nailed to the cross with Christ and buried in the baptism. In Christ I am a totally new creation, not guilty of what the old me have done.

Therefore, if anyone is in Christ, he is a new creation. The old has passed away; behold, the new has come.
2. Corinthians 5:17

The new creation that I now am, is not guilty in any sins that I have done. And because the sacrifice Jesus did was an eternal sacrifice, my new creation that right now is dwelling in corrupted flesh, will remain holy until I get my new body in the resurrection; even though I still might sin.

He has no need, like those high priests, to offer sacrifices daily, first for his own sins and then for those of the people, since he did this once for all when he offered up himself.
Hebrews 7:27

...he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption.
Hebrews 9:12

And by that will we have been sanctified through the offering of the body of Jesus Christ once for all… But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God… Therefore, brothers, since we have confidence to enter the holy places by the blood of Jesus
Hebrews 10:10, 12, 19

To be righteous means to have a right standing before God. It means that when you stand before God, you don’t have to feel embarrassed because you have not been able to live up to His standards. To be justified mean that the charges that was brought against you have been dropped.

You are because of the redemptive work of Jesus Christ both righteous and justified and can now come boldly before God.