Jewish Christians (sometimes called also "Hebrew Christians" or "Christian Jews") is a term which can have two meanings, a historical one and a contemporary one. The historical term refers to early Christians of or attracted to Jewish culture. This concept deals with the relation between the traditional ethnic religious beliefs and practices of Judaism (including Jewish proselytes) and the then-emergent universal religious concepts of Hellenistic Judaism and then Christianity. The contemporary concept simply refers to individuals of certain Jewish ancestry or heritage, who is an adherent of some form of Christianity and not Judaism. This includes "converts" from Judaism to Christianity and ethnic Jews who for one reason or another had not been indoctrinated into Judaism.
Jesus, his Twelve Apostles, the Elders, his family, and essentially all of his early followers were Jewish or Jewish Proselytes.
The term "Early Jewish Christians" is often used in discussing the Early History of Christianity, see also Early Christianity and History of early Christianity. Hence the 3,000 converts on Pentecost (Sivan 6), following the death and resurrection of Jesus (Nisan 14 or 15), described in Acts of the Apostles 2, were all Jews and Proselytes. Samaritans were not Jewish (Judean), but are still identified with the tribes of Israel and also numbered among the early followers, as is the Ethiopian eunuch (Acts 8)[2]. Traditionally the Roman Centurion Cornelius is considered the first Gentile convert, as recorded in Acts 10, albeit he too is a "God-fearer" proselyte who participated in a Jewish synagogue. The major division prior to that time was between Hellenistic and non-Hellenistic Jews or Koine Greek (Acts 6) and Aramaic (Acts 1:19) speakers. The conversion and acceptance of the Gentile Cornelius can be described in terms of the Judaic teaching which describes strangers becoming part of the community (Isaiah 56:3-7). Acts does not use the term "Jewish Christians", rather those led by James the Just, Simon Peter, and John the Apostle, the "Pillars of the Church", were called followers of "The Way". Later groups, or perhaps the same group by different names, were the Ebionites and Elkasites.
The "Christian" appellation was first applied to the followers after Paul of Tarsus started preaching at Antioch (Acts 11:25-26). Paul made explicit in Galatians 1:7 that he did not discuss with the "Pillars of the Church" after he had received his revelation to be an apostle (1:15-16), that he saw no one except Cephas (Peter) and James, when he was in Jerusalem three years after the revelation (1:18-24) and implies he did not explain his gospel to them until 14 years later (2:1-2) in a subsequent trip to Jerusalem The division between those who followed Mosaic law and were circumcised and those who were not circumcised was highlighted in his Epistle to the Galatians 2:7-9:
"On the contrary, when they saw that I had been entrusted with the gospel for the uncircumcised, just as Peter had been entrusted with the gospel for the circumcised (for he who worked through Peter making him an apostle to the circumcised also worked through me in sending me to the Gentiles), and when James and Cephas and John, who were acknowledged pillars, recognized the grace that had been given to me, they gave to Barnabas and me the right hand of fellowship, agreeing that we should go to the Gentiles and they to the circumcised." (NRSV) These terms (circumcised/uncircumcised) are generally interpreted to mean Jews and Greeks respectively, who were predominant in the region; however this is an oversimplification as 1st century Iudaea Province also had some Jews who no longer circumcised (sometimes called Hellenized Jews), and some Greeks (called Proselytes or Judaizers) and others such as Egyptians, Ethiopians, and Arabs who did. See also Abrahamic religion and Circumcision controversy in early Christianity#Jewish background.
Jesus is frequently called the "Nazarene" (Matthew 2:23; Mark 10:47; 14:67; 16:6; Luke 24:19; John 18:5; 18:7; 19:19; Acts 2:22; 3:6; 4:10; 6:14; 22:8. Named after him, the followers of Paul are the Nazarenes (Acts 24:5, Jerome, Commentary on Isaiah 9:1). The Council of Jerusalem, according to Acts 15, determined that circumcision was not required of Gentile converts, only avoidance of "pollution of idols, fornication, things strangled, and blood" (KJV, Acts 15:20). The basis for these prohibitions is unclear, Acts 15:21 states only: "For Moses of old time hath in every city them that preach him, being read in the synagogues every sabbath day", the implication being that they are based on the Law of Moses. Many, beginning with Augustine of Hippo[6] consider them to be based on the Noahide Laws, while some modern scholars reject the connection to Noahide Law (Genesis 9) and instead see Lev 17-18 as the basis. Also unclear is whether this meant that this Law in some way applied to them or merely that the requirements were imposed to facilitate common participation in the Christian community by Gentiles who would be in constant relation with the Jewish Christians who would be constantly reminded of their obligation to follow the Law. See also Biblical law in Christianity and Expounding of the Law. The early Jewish Christians included those who believed non-Jews must become Jews and adopt Jewish customs. They were derogatively called Judaizers, and even Paul used this term against Jesus's student Peter in public according to Young's Literal Translation of Gal 2:14:

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But when I saw that they are not walking uprightly to the truth of the good news, I said to Peter before all, `If thou, being a Jew, in the manner of the nations dost live, and not in the manner of the Jews, how the nations dost thou compel to Judaize?

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However, Barnabas, Paul's partner up till then, sided with Peter (Gal 2:13, Acts 15:39-40). Catholic Encyclopedia: Judaizers: The Incident at Antioch claims: "St. Paul's account of the incident leaves no doubt that St. Peter saw the justice of the rebuke." however, L. Michael White's From Jesus to Christianity
claims: "The blowup with Peter was a total failure of political bravado, and Paul soon left Antioch as persona non grata, never again to return." See also Pauline Christianity. Scholar James D. G. Dunn, who coined the phrase New Perspective on Paul, has proposed that Peter was the bridge-man (i.e. the pontifex maximus) between the two other "prominent leading figures" of early Christianity: Paul and James the Just. Marcion in the 2nd century, called the "most dangerous" heretic, rejected the Twelve Apostles, and interpreted a Jesus who rejected the Law of Moses using 10 Pauline Epistles and the Gospel of Luke. For example, his version of Luke 23:2 : "We found this fellow [Jesus] perverting the nation and destroying the law and the prophets". Irenaeus in turn rejected Marcion and praised the Twelve Apostles in his Against Heresies 3.12.12: "...being brought over to the doctrine of Simon Magus, they have apostatized in their opinions from Him who is God, and imagined that they have themselves discovered more than the apostles, by finding out another god; and [maintained] that the apostles preached the Gospel still somewhat under the influence of Jewish opinions, but that they themselves are purer [in doctrine], and more intelligent, than the apostles." According to Eusebius' History of the Church 4.5.3-4: the first 15 Bishops of Jerusalem were "of the circumcision". The Romans destroyed the Jewish leadership in Jerusalem in year 135 during the Bar Kokhba Revolt. However, that doesn't necessarily mean an end to Jewish Christianity, any more than Valerian's Massacre of 258, (when he killed all Christian bishops, presbyters, and deacons, including Pope Sixtus II and Antipope Novatian and Cyprian of Carthage), meant an end to Roman Christianity.