Ioudaios

This is the core use of the word: a person who belongs to the Jewish people. By the time of the New Testament the name had widened from "someone from the tribe or territory of Judah" to cover the whole nation of Israel as the world then knew it, defined less by where a person lived than by descent, the worship of the one God, circumcision, and life under the law. So a Jew might be born in Jerusalem, Tarsus, or Alexandria and still be called by this one name.

The word almost always appears beside its opposite. Again and again the New Testament pairs "Jew" with "Greek" or with "the nations" to map out the whole human race in two halves: the people who already had God's covenant and everyone else. Paul leans on this pairing constantly, both to insist that the good news came "to the Jew first" and to argue that in Christ the old dividing line no longer decides who belongs to God.

One special use sits inside this meaning rather than apart from it. The Gospel of John repeatedly says "the Jews" did or said something, and there it very often points not to the people as a whole but to a specific group within Jerusalem, the leaders and authorities who opposed Jesus. This is a narrowing of the same word to a hostile faction, not a separate dictionary sense, which is why it sits here as a sub-use that readers must weigh verse by verse.

Alongside naming a person, the same word works as a describing word: Jewish. It marks something as belonging to the Jewish world rather than identifying a particular individual. The New Testament uses it this way for a region, for a woman's background, and for religious practice.

This is the same word and the same underlying idea as "a Jew"; the difference is only that here it qualifies a thing or a custom instead of pointing to a person. English keeps the two apart with separate words, "Jew" and "Jewish," which is why it helps to treat them as distinct uses even though Greek uses one form for both.