In Esther 8:17, how can one become a Jew?
"And in every province and city, wherever the king's command and decree came, the Jews had joy and gladness, a feast and a holiday. Then many of the people of the land became Jews, because fear of the Jews fell upon them" (Esther 8:17).
The most common way was to be born to an Israelite, and if you were male, to be circumcised.
But God did not limit His people to just those descended from Jacob. "Thus says the LORD of hosts: 'Peoples shall yet come, Inhabitants of many cities; The inhabitants of one city shall go to another, saying, "Let us continue to go and pray before the LORD, And seek the LORD of hosts. I myself will go also." Yes, many peoples and strong nations Shall come to seek the LORD of hosts in Jerusalem, And to pray before the LORD.' Thus says the LORD of hosts: 'In those days ten men from every language of the nations shall grasp the sleeve of a Jewish man, saying, "Let us go with you, for we have heard that God is with you."'" (Zechariah 8:20-23).
Even at the beginning, Abraham was instructed to bring under the covenant those in his household who were slaves. "He who is eight days old among you shall be circumcised, every male child in your generations, he who is born in your house or bought with money from any foreigner who is not your descendant. He who is born in your house and he who is bought with your money must be circumcised, and My covenant shall be in your flesh for an everlasting covenant" (Genesis 17:12-13). This law is reflected in the Law of Moses: "And when a stranger dwells with you and wants to keep the Passover to the LORD, let all his males be circumcised, and then let him come near and keep it; and he shall be as a native of the land. For no uncircumcised person shall eat it" (Exodus 12:48).
Other mentions of converts are in Isaiah 56:3 and Nehemiah 10:28
In the New Testament, there is mention of Jewish proselytes (converts) (Matthew 23:15; Acts 2:10; 13:43).
Ruth is probably the most famous of the Gentiles who became an Israelite. "But Ruth said: 'Entreat me not to leave you, Or to turn back from following after you; For wherever you go, I will go; And wherever you lodge, I will lodge; Your people shall be my people, And your God, my God. Where you die, I will die, And there will I be buried. The LORD do so to me, and more also, If anything but death parts you and me'" (Ruth 1:16-17). Basically it requires the person to take on the Mosaical covenant and follow all aspects of it, just as any Israelite was expected to do.