Thank you for your question.
In the Old Testament the term Jew is applied to those belonging to the Kingdom of Judah.
Using Strong’s Concordance:
Jew(s) 03064 Yahuwdiy; patronymically from 03063
Translated in the AV as - Jew 74 times, Jew + 0376 1, Judah 1; 76
03063 Yahuwdah from 03034 yadah – ‘praise’
Translated in the AV as - Judah 808 times
1) The son of Jacob by Leah
2) The tribe descended from Judah the son of Jacob
3) The territory occupied by the tribe of Judah
4) The kingdom comprised of the tribes of Judah and Benjamin which occupied the southern part of Canaan after the nation split upon the death of Solomon.
Probably the best definition of a Jew is one (or descendant of one) who was part of the kingdom of Judah after the ten tribes split away following the death of Solomon. In practice it would include Judah, Benjamin, those of Levi who continued to officiate at the Temple and a sprinkling of the other tribes who moved to Judea to avoid worshipping the Golden calves.
The word is first used in:
2 Kings 16:6 At that time Rezin king of Syria recovered Elath to Syria, and drave the Jews <03064> from Elath...
In Jeremiah ‘Jew’ and ‘Hebrew’ are shown to be synonymous terms:
Jeremiah 34:9 That every man should let his manservant, and every man his maidservant, being an Hebrew or an Hebrewess, go free; that none should serve himself of them, to wit, of a Jew <03064> his brother.
Jew is used to describe those carried away captive from Jerusalem and the Kingdom of Judah by Nebuchadrezzar.
Jeremiah 52:28 This is the people whom Nebuchadrezzar carried away captive: in the seventh year three thousand Jews <03064> and three and twenty:
30 In the three and twentieth year of Nebuchadrezzar Nebuzaradan the captain of the guard carried away captive of the Jews <03064> seven hundred forty and five persons: all the persons were four thousand and six hundred.
In Esther it is used collectively for those of the captivity of Judah scattered throughout the Persian Empire.
Esther 3:6 And he thought scorn to lay hands on Mordecai alone; for they had shewed him the people of Mordecai: wherefore Haman sought to destroy all the Jews <03064> that were throughout the whole kingdom of Ahasuerus, even the people of Mordecai.
On their deliverance, the feast of Purim (lots – from the casting of lots by Haman) was instituted and became one of the distinguishing celebrations of Jews. It is still observed by Jews today.
Esther 9:27 The Jews <03064> ordained, and took upon them, and upon their seed, and upon all such as joined themselves unto them, so as it should not fail, that they would keep these two days according to their writing, and according to their appointed time every year;
28 And that these days should be remembered and kept throughout every generation, every family, every province, and every city; and that these days of Purim should not fail from among the Jews <03064>, nor the memorial of them perish from their seed.
In Nehemiah it is used collectively for those (including Levites) who returned from captivity in Babylon to live in Judea and rebuild Jerusalem.
Nehemiah 4:2 And he spake before his brethren and the army of Samaria, and said, What do these feeble Jews <03064>? will they fortify themselves? will they sacrifice? will they make an end in a day? will they revive the stones out of the heaps of the rubbish which are burned?
That Jews will be restored to God’s favour in the time to come is confirmed by Zechariah’s prophecy.
Zechariah 8:23 Thus saith the LORD of hosts; In those days it shall come to pass, that ten men shall take hold out of all languages of the nations, even shall take hold of the skirt of him that is a Jew <03064>, saying, We will go with you: for we have heard that God is with you.
In the New Testament the term Jew is applied to those belonging to the Kingdom of Judah who lived not only in Judea but also in Galilee and in dispersion throughout the Roman world. The common bond was the Law of Moses and the Temple at Jerusalem. Their cohesion was maintained by Synagogues wherever they lived, Sabbath assembly and the reading of the Old Testament Scripture.
Using Strong’s Concordance:
Jew(s) 2453 Ioudaios from 2448 (Iouda – Judah)
Translated in the AV as - Jew 193 times, of Judea 1, Jewess 2; 196
1) Jewish, belonging to the Jewish nation
2) Jewish as respects to birth, origin, religion
It first appears in Matthew:
Matthew 2:2 Saying, Where is he that is born King of the Jews <2453>? for we have seen his star in the east, and are come to worship him.
The next time it appears is when Jesus confirms that he is the king of the Jews.
Matthew 27:11 And Jesus stood before the governor: and the governor asked him, saying, Art thou the King of the Jews <2453>? And Jesus said unto him, Thou sayest.
In Acts we see that at Pentecost Jews had come to Jerusalem from every nation where they spoke different languages.
Acts 2:5 And there were dwelling at Jerusalem Jews <2453>, devout men, out of every nation under heaven.
The Gospel was preached first to the Jews only.
Matthew 10:5 These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not:
6 But go rather to the lost sheep of the house of Israel.
The disciples at first preached to the Jews only.
Acts 11:19 Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews <2453> only.
The prescribed order was the Jew first, then Samaria (Samaritans) then the Gentiles.
Acts 1:8 But ye shall receive power, after that the Holy Spirit is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth.
Philip extended the preaching to the Samaritans (Acts 8:5).
Peter extended the preaching to the Gentiles (Acts 10:34).
Paul, of course, described himself as the apostle to the Gentiles (Romans 11:13).
The real Jews are Hebrews, descendants of Abraham. As such they are the natural heirs to the promises made to Abraham.
Acts 3:25 Ye are the children of the prophets, and of the covenant which God made with our fathers, saying unto Abraham, And in thy seed shall all the kindreds of the earth be blessed.
Genesis 17:7 And I will establish my covenant between me and thee and thy seed after thee in their generations for an everlasting covenant, to be a God unto thee, and to thy seed after thee.
8 And I will give unto thee, and to thy seed after thee, the land wherein thou art a stranger, all the land of Canaan, for an everlasting possession; and I will be their God.
Genesis 22:18 And in thy seed shall all the nations of the earth be blessed; because thou hast obeyed my voice.
This seed of Abraham in whom all nations will be blessed is of course Christ.
Galatians 3:16 Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ.
Because the Jews rejected and killed their Messiah (although a large number subsequently accepted him) Jerusalem was destroyed and they were again dispersed among the nations.
The gospel preached to the Gentiles requires them to be baptised into Christ. This act makes them spiritual Jews and heirs of the promises made to Abraham.
Galatians 3:27 For as many of you as have been baptized into Christ have put on Christ.
28 There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus.
29 And if ye be Christ’s, then are ye Abraham’s seed, and heirs according to the promise.
A spiritual Jew is one circumcised in heart and not in the flesh.
Romans 2:28 For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:
29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.
Natural Jews cannot assimilate into the nations among whom they live. Since AD 70 they have suffered persecution in every nation (In fulfilment of Moses’ prophecy in Deuteronomy 28:63-67). Although now back in their land a final siege and dispersion before their deliverance is prophesied in Zechariah 14 and Ezekiel 38.
Thus God has not cast them off forever. When the fullness of the Gentiles is brought in at the return of Christ, the kingdom will be restored to Israel and Christ will rule from Jerusalem.
Romans 11:25 For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in.
26 And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob:
27 For this is my covenant unto them, when I shall take away their sins.
Zechariah 12:10 And I will pour upon the house of David, and upon the inhabitants of Jerusalem, the spirit of grace and of supplications: and they shall look upon me whom they have pierced, and they shall mourn for him, as one mourneth for his only son, and shall be in bitterness for him, as one that is in bitterness for his firstborn.
Luke 1:31 And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS.
32 He shall be great, and shall be called the Son of the Highest: and the Lord God shall give unto him the throne of his father David:
33 And he shall reign over the house of Jacob for ever; and of his kingdom there shall be no end.
In conclusion real Jews are descendants of Abraham who originated from Judeah.
Spiritual Jews are those baptised into Christ to become heirs of the promises made to Abraham.
The fulfilment of these promises requires resurrection from the dead to eternal life on this earth. This is the gospel of the kingdom of God.
I hope this helps