What is justification? It's not a word Scripture uses a lot. While not exclusively a Pauline word, it's primarily Paul's. What does it mean?

A person is justified when he (or she) is declared to be just. What does that mean with respect to Scripture? Let's consider a few places where Scripture mentions justification:

If there be a controversy between men, and they resort to judgment, and they judge their case; then they shall justify the righteous, and condemn the wicked.
Deuteronomy 25:1, DBY

Then was kindled the anger of Elihu the son of Barachel, the Buzite, of the family of Ram: against Job was his anger kindled, because he justified himself rather than God;
Job 32:2, DBY

And enter not into judgment with thy servant; for in thy sight no man living shall be justified.
Psalm 143:2, DBY

Be it known unto you, therefore, brethren, that through this man remission of sins is preached to you, and from all things from which ye could not be justified in the law of Moses, in him every one that believes is justified.
Acts 13:38--39, DBY

to him who does not work, but believes on him who justifies the ungodly, his faith is reckoned as righteousness
Romans 4:5, DBY

The passage in Deuteronomy establishes what it means to justify: when the judge looks into a case, he can either condemn or justify. In other words, he can either pronounce the defendant guilty or not guilty.  The Law said the judge must justify the righteous (that is, acknowledge the righteousness of the righteous) or condemn the wicked (that, acknowledge the wickedness of the wicked).  The Law expressly  forbids justifying the wicked, just like it forbids condemning the righteous.

Psalm 143 further develops this thought: it establishes that there is no hope for anyone to be justified in God's sight. Why? Because God must condemn the wicked just like He must justify the righteous. There is no such thing as a righteous man, so there's no man God can justify if He were to enter into judgment with them.  So the Psalmist asks the Lord not to enter into judgment with him: there is no hope for a man with whom God enters into judgment.

That's not strictly true, is it? There is precisely one Man that is righteous. There is exactly one Man that God can honestly look at and justify. And this is the foundation of the entire doctrine of justification. 

Paul takes up the thread of justification in Acts 13. There he is addressing the synagogue at Antioch of Pisidia.  What does he say to them? He tells them that under Moses' Law there were any number of sins from which there is no justification. But, he tells them, through "this Man" there is remission of sins: and it's such a complete remission that the one who believes is justified "from all things."  That is an important statement: imagine what the synagogue must have thought at hearing that!  Unlimited remission of sins, regardless of the crime.

What is justification? It is the judge saying "Not Guilty!"  The Psalmist said it's not possible for God to justify if He enters into judgment with any living man. But Paul says something different: through "this Man" there is unlimited remission of sins.

How can that be?  The answer is in Romans 3:

21 But now without law righteousness of God is manifested, borne witness to by the law and the prophets;  22 righteousness of God by faith of Jesus Christ towards all, and upon all those who believe: for there is no difference;  23 for all have sinned, and come short of the glory of God;  24 being justified freely by his grace through the redemption which is in Christ Jesus;  25 whom God has set forth a mercy-seat, through faith in his blood, for the shewing forth of his righteousness, in respect of the passing by the sins that had taken place before, through the forbearance of God; 26 for the shewing forth of his righteousness in the present time, so that he should be just, and justify him that is of the faith of Jesus.
Romans 3:21–26, DBY

The righteousness of God is manifested (that is, "revealed") now, without law. But this without-law righteousness isn't new, it's just newly revealed. In fact, the Law and the prophets bore witness to this righteousness (Romans 3:21).  What is this righteousness? It's the righteousness of God. It's not a righteousness for God, it's the righteousness of God.  And this righteousness is upon all who believe (Romans 3:22).  God justifies freely by His grace. This is the most important idea in the whole Gospel: God justifies by His grace.  It's not God saying "Not Guilty!" because we're not guilty: it's God saying "Not Guilty!" by His grace. It's His declaring someone to be righteous because that's what He wants to do, not regarding the plain truth that the person is very, very guilty. What gives God the right to declare someone not guilty who is indeed guilty? How can God do what He commands we must never do? That's the second half of Romans 3:24. God justifies freely, by His grace, through the redemption in Christ Jesus. How did this redemption come to be in Christ Jesus? Because of His blood (Romans 3:25). Christ Jesus has died for us, and that is why God can justify.  Romans 3:26 puts the whole passage– indeed the whole epistle– into perspective: God wants to justify, but He must do so justly. God wants to look at guilty sinners and say, "Not Guilty!" But if God just covers up our sins, He doesn't remain just. If a man helps cover up a crime, he becomes an accomplice after the fact. God can't become an accomplice to our sins by just ignoring them, He must be righteous in how He deals with us.

And so Romans 3:26 answers the question: can God justify sinners and still be just Himself?  Yes, He can. He does so, not by just ignoring our sins, but by judging them in Christ.  The Son of God suffered and died in our place, so we can be justified by God. 

That's not the whole story. God justifies freely, by His grace, because Christ has died in the place of guilty sinners. God declares guilty sinners "Not Guilty!" because He has already judged the sins those wicked sinners committed. He judged them on His own Son.   But how can we get in on that deal? I know I'm guilty, how can I get a piece of this freely-by-His-grace justifying? That's what's answered in Romans 4.

Romans 4 takes up the question of how we can be justified. Romans 3 tells us how God can justify, Romans 4 tells us how we can get a piece of it. Verse 5 is the executive summary, so to speak:

to him who does not work, but believes on him who justifies the ungodly, his faith is reckoned as righteousness
Romans 4:5, DBY

God doesn't justify the one who works. God justifies the one who believes. Let's consider that a moment: Scripture doesn't give us anything to do. It doesn't tell us we need to ask Jesus into our hearts; it doesn't say we need to receive Christ; it doesn't say we need to pray or be remorseful, or even repent. It says that God justifies the one who believes. 

So justification is God declaring we aren't guilty. God justifies justly, because Christ Jesus' blood has been shed for us.  God doesn't justify the one who works: He justifies the one who believes. God doesn't look for anything on our part, except that we believe Him.

We notice that justification doesn't imply anything at all about the one who is justified. If someone believes God, and God declares him (or her) "Not Guilty", then we don't expect anything has happened in that person. We don't expect the person is actually changed: in fact, we expect that person hasn't changed in any way, because Romans 4:5 specifically tells us that God justifies the ungodly. So when an ungodly person believes, and God justifies that ungodly person, that person is still ungodly. God's declaring "Not Guilty!" doesn't mean the person didn't actually do the crime. It doesn't mean that person won't do more crimes, or even the same crimes over and over. What it means is, God won't hold them against that person. 

Do we have Scripture for that? Of course we do! Romans 4 continues:  

7 Blessed they whose lawlessnesses have been forgiven, and whose sins have been covered: 8 blessed the man to whom the Lord shall not at all reckon sin.
Romans 4:7–8, DBY

The man (or woman) who believes is the man (or woman) God justifies. That man (or woman) is now one to whom God will "not at all reckon sin". That means, not only has God said "Not Guilty!", but He won't consider any sins we commit– in the past or future– as being ours. So when an ungodly person believes, God justifies that person and declares that person to be righteous, but He also says that He will never consider any sins as belonging to that person, regardless of what that person does.

This is what Paul told the synagogue at Antioch of Pisidia, isn't it? God justifies the one who believes from all things. 

But it's important to remember this central fact: justification implies no change at all in the justified one. When God justifies a believing sinner, the believing sinner is not at all changed by his (or her) justification. 

Justification in and of itself doesn't imply new life, nor a home in Heaven. What it implies is, God has looked at a sinner and declared that sinner not guilty. It implies God will never, for any reason, look at that person and see sins. Justification is what brings us peace with God.

Therefore having been justified on the principle of faith, we have peace towards God through our Lord Jesus Christ;
Romans 5:1, DBY