A Judaizer is one that believes in Jesus, but also believes that you must follow the law of Moses.
There were many Judaizers in the early church. The apostle James was one of them. Many did not understand the Gospel and justification by faith apart from the works of the law. Paul was the only apostle that fully understood that Christians were no longer required to follow the law. The reason for this is because Paul received the Gospel directly from Jesus Christ, Galatians 1:11,12. It is the Gospel that frees us from the law of Moses.
James sent men to spy on Peter and Barnabas to see if they were eating with Gentiles, Galatians 2:12. This was a direct violation of the law of Moses. When the Jews walked into the dinner there at the church in Antioch, Peter and Barnabas went under the table. Paul saw the whole thing and gave everyone a good tongue lashing, Galatians 2:14-21.
Later on in Acts 15, Paul wanted to introduce some Gentile believers to the church there in Jerusalem. When James and some of the others got word that they were coming they said, "Except that you are circumcised after the manner of Moses, you cannot be saved" Acts 15:1. This caused some dispute. If it had not been for all of the things that God was doing with the Gentiles through Paul they probably would have circumcised them, but because of Paul's testimony they let them off.
WHY CHRISTIANS ARE NOT UNDER THE LAW OF MOSES OR ANY LAWS.
All through the Old Testament and into the New Testament God's people are called to live by faith and not by rules, laws or religion. Paul wrote, "The Just Shall Live By Faith" Romans 1:17. Living by rules, laws and religion is not living by faith. While Jesus was dying on the cross God tore the veil that covered the "Holy of Holies" from the top to the bottom, Matthew 27:51. This signified the end of the Old Covenant, the law and the Jewish religion. Many in the early church could not grasp the meaning of this event and became Judaizers. Many today cannot believe that Christians are free from the law and try to keep the commandments.
Living by laws, rules and commandments is a denial of the Gospel.
Jesus is the end and the fulfillment of the Old Covenant, the law and the Jewish religion. Paul wrote, "But now the righteousness of God WITHOUT THE LAW is manifested, being witnessed by the law and the prophets" Romans 3:21. If you want to see the righteousness of God, don't look to the law, Jesus is God's law incarnate in human flesh. Jesus is the New Covenant. The Old Covenant and the law have passed away, Hebrews 8:13. The new and living way is by faith in Christ and his Gospel, which brings with it the indwelling Holy Spirit.
Some of the Galatians were trying to receive the Holy Spirit by the works of the law, this is what Paul said to them, "This only would I learn of you, Did you receive the Holy Spirit by the works of the law (because you did something) or by the hearing of faith?" (hearing and believing the Gospel) Galatians 3:2. Many do not have the Spirit because they are trying to obtain it by the works of the law instead of by faith in Christ and his Gospel.