The short answer is that Abraham was a Gentile who was chosen by God to be the ancestor of the Jewish people.
The term Jew is a shortened form of the word Judah, which was the name of one of the tribes of Israel, the tribe of David and of Jesus (Matthew 1:1). Judah was also the name of the southern half of the kingdom of Israel when it split into two parts (1 Kings 12), because it was dominated by the large tribe of Judah. The first time the word Jew is used in the Bible is during the exile (2 Kings 25:25), and may have been a term coined by the Babylonians or Persians to refer to the people in their midst who had come from the kingdom of Judah. By New Testament times, Jew was a common term, and it has remained in usage unto this day. Obviously, Abraham was never referred to by this term.
The term Gentile is simply from the Latin word for “nation.” When the term Gentiles is used in Scripture, it means “the nations.” If a person is a Gentile, he or she is a member of one of the many nations in the world. When Jew and Gentile are juxtaposed, the contrast is between one who is a member of God’s chosen nation and one who is a member of one of the hundreds of other nations or ethnic groups not chosen for special blessing in the way that Israel was. In this context, Gentile simply means “not Jewish.”
Abraham started out as a member of one of the many nations or ethnic groups that were in existence at the time. (Of course, these are somewhat artificial distinctions, because all people came from Adam and Eve, and all people are related if the genealogy is traced back far enough. However, after Babel, people started to segregate into smaller groups and develop into distinctive ethnic groups, clans, and nations.) Abraham was living in Ur of the Chaldeans. While he was there, God called him:
“The Lord had said to Abram, ‘Go from your country, your people and your father’s household to the land I will show you. I will make you into a great nation, and I will bless you; I will make your name great, and you will be a blessing. I will bless those who bless you, and whoever curses you I will curse; and all peoples on earth will be blessed through you’” (Genesis 12:1–3).
At that point, Abraham was called and set apart for God’s purposes. He left his native country and people behind. His descendants would become the great nation of Israel, God’s chosen people, but the line had not been narrowed down yet. Abraham had several sons: Isaac, by his wife Sarah; Ishmael, by a slave of Sarah; and other sons with another wife after Sarah died. However, it was only Isaac who was the chosen one. Isaac had two sons, Jacob and Esau, and only Jacob (whose name was later changed to Israel) was the chosen one. All of Jacob’s 12 sons were included as patriarchs in the nation of Israel and became the basis for the 12 tribes. It was at this point that the line had been finally set; however, the wives of Jacob’s sons were still from outside the family.
With the next generation of Jacob’s grandchildren, the descendants of Jacob became a distinct people group, and they were distinguished from the Egyptians among whom they were living as having all descended from one man named Israel. By the time of Exodus 1, they were recognized as a distinct nation.
Technically, no one was called a “Jew” before the exile; however, the people who became known as Jews were a distinct ethnic group by the time of Exodus 1. They were a distinct clan by the time of Jacob and his sons. Abraham was a Gentile, that is, a member of one of the many nations that had developed by his time. The Jews came from Abraham because he was chosen by God from among the nations to be the origin of a new nation. The Jews of Jesus’ day looked to Abraham (not Jacob/Israel) as the head of their race (see Matthew 3:9; Luke 3:8; John 8:39; Acts 13:26; Galatians 3:7). If one is thinking in these terms, it would not be wrong to think of Abraham as “the first Jew,” although that’s not technically correct.