Romans 3:1 What advantage then hath the Jew? or what profit is there of circumcision?
Rom 3:2 Much every way: chiefly, because that unto them were committed the oracles of God.

The Jewish people have an advantage and were also considered to be the elect of God because they were chosen first to receive the law of God which is holy and bring forth the way of salvation to all mankind. This was done through their seed in succession until the coming of the Lord Jesus Christ. Even though the nation of Israel as a whole did not recognize Him when He came, that did not change the fact that He was indeed the one Moses and the prophets promised and prophesied that would come. He did come as it was written of Him.

The Jews were chosen and elected to bring forth the Messiah from out of their midst. He did not come the first time as they assumed to set up His earthly kingdom. That stumped them and caused them to enter into unbelief. He will set up His literal earthly kingdom at His second coming but the 1st coming was to make an offering for sin.

The blood of goats, bulls, and lambs cannot take away sin but only pointed to the one to come who had the power to take away sins. Those who offered those sacrifices in faith of that which was to come were saved by faith along with us who believe in Jesus at this present time.

He came the first time to die for and pay for our sins so that all who receive Him can be saved from sin and inherit everlasting life.

Rom 3:3 For what if some did not believe? shall their unbelief make the faith of God without effect?
Rom 3:4 God forbid: yea, let God be true, but every man a liar; as it is written, That thou mightest be justified in thy sayings, and mightest overcome when thou art judged.

Our unbelief or the unbelief of the Jewish people does not nullify the word of God.

There are people today who do not believe in a literal place called hell. Just because they do not want to believe that does not make it any less real. Our unbelief can never change the truth of God's word but it can hinder us from the benefits that come when we believe in God's word.

Rom 3:5 But if our unrighteousness commend the righteousness of God, what shall we say? Is God unrighteous who taketh vengeance? I (speak as a man)
Rom 3:6 God forbid: for then how shall God judge the world?

The Jews were unrighteous in that they did not believe. Because of their unbelief the gospel went to the Gentiles who believed. Paul is asking here if God is justified to bring judgment since that was the plan and it was meant to be.

Rom 3:7 For if the truth of God hath more abounded through my lie unto his glory; why yet am I also judged as a sinner?
Rom 3:8 And not rather, (as we be slanderously reported, and as some affirm that we say,) Let us do evil, that good may come? whose damnation is just.

Paul is trying to make it clear that all will be judged including unbelieving Jews even though their sin of unbelief brought good in that it brought the gospel to the Gentiles. As a nation they rejected Jesus but the individuals within that nation, they will still be judged whether or not they believed as individuals. The apostles and many other disciples believed, so they are without excuse.

Rom 3:9 What then? are we better than they? No, in no wise: for we have before proved both Jews and Gentiles, that they are all under sin;
Rom 3:10 As it is written, There is none righteous, no, not one:

This discounts the modern teaching that says there is one way of salvation for Jews and another way for Gentiles. The bible never teaches such a doctrine. All are under sin and all will have to be saved from sin by faith in Jesus Christ to enter into eternal life.

Rom 3:11 There is none that understandeth, there is none that seeketh after God.
Rom 3:12 They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.
Rom 3:13 Their throat is an open sepulcher; with their tongues they have used deceit; the poison of asps is under their lips:
Rom 3:14 Whose mouth is full of cursing and bitterness:
Rom 3:15 Their feet are swift to shed blood:
Rom 3:16 Destruction and misery are in their ways:
Rom 3:17 And the way of peace have they not known:
Rom 3:18 There is no fear of God before their eyes.
Rom 3:19 Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God.

All are guilty before God. All of us are on our way to hell and an eternity without God. God made a way out and if we neglect that way, we are on our way to eternity without God in a place called hell.

Rom 3:20 Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin.

The law shows us sin and condemns us. It shows that we all have sinned. We cannot be justified by trying to keep the law. The law shows us that we are not justified. Both Jew and Gentile is eternally lost because the law shows us that we have sinned and the wages of sin is eternal death in hell.

Rom 3:21 But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets;

Since there is no way we can keep the whole law flawlessly, God gave us someone who could keep it and not sin in any point. This had to be a man because it was man that brought sin into the world at the beginning.  A perfect man had to come and not sin in any point of the law. Yet a mere man could not live on earth without breaking the law sooner or later. Jesus was the Son of God and became flesh or man to live among us, keep the law in all points and yet die for breaking the law even though He didn't break it.  He had the penalty or judgment of death upon Him even though He did not deserve it for any sin He committed Himself.  This death can count on our behalf when we repent of sin and believe upon Him as savior from this sin.  He took our punishment for us but it only counts for those who believe in Him.  His death does not count for those who reject Him and refuse to turn from sin. 

Rom 3:22 Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference:
Rom 3:23 For all have sinned, and come short of the glory of God;
Rom 3:24 Being justified freely by his grace through the redemption that is in Christ Jesus:
Rom 3:25 Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;
Rom 3:26 To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.

Because of the sacrifice of Jesus, we are cleansed from sin by the blood He shed. We are free from sin just by believing what He did and receiving Him by faith. If we reject this only way of salvation from sin, we remain in our sin and condemned. To refuse and not believe and continue that way until the time we die physically, is to miss salvation. We will die in sin and be that way forever.

God made a perfect way that we can be free from sin and be filled with His spirit. We do not have to go it alone and use will power. We can be justified or made just as if we had never sinned and then be filled with the Holy Spirit which will live inside of us so that we can have a helper, a teacher, and a comforter always with us to keep us from returning to the sin Jesus' death saved us from.

When He was resurrected from the dead, it showed what will happen to those who believe in Him. We also will be raised from the dead never to die again.

Rom 3:27 Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith.
Rom 3:28 Therefore we conclude that a man is justified by faith without the deeds of the law.

This is not teaching us that we can return to sin and still be saved. The deeds of the law are the rituals, ordinances and sacrifices that were put in place until the final sacrifice came which is the Son of God, Jesus Christ.

Faith now saves us without doing all those rituals, ceremonies and sacrifices. Jesus fulfilled that part of the law. We still are under the moral righteous law that is repeated in the New Testament.

Rom 3:29 Is he the God of the Jews only? is he not also of the Gentiles? Yes, of the Gentiles also:
Rom 3:30 Seeing it is one God, which shall justify the circumcision by faith, and uncircumcision through faith.
Rom 3:31 Do we then make void the law through faith? God forbid: yea, we establish the law.

Jesus completed and fulfilled the letter of the law. He made the complete sacrifice for all our sins. His sacrifice is sufficient to save all mankind but only those that turn from sin and are redeemed by His blood are forgiven and given the Holy Spirit.

Unbelief is a killer. If we don't believe to the point of turning from sin and turning to Jesus for salvation, there is no other way to be saved.