What does Jewish mean?

their mother is Jewish or they convert to the Jewish religion

If a person doesn't fall into one of these two groups they are not Jewish. Jews are part of the religion Judaism, but Jews are also a family and a people. The Jewish homeland is Israel.

To clarify: There is no such thing as being 'ethnically jewish'. Jews come from all ethnic backgrounds. Nor is there one unifying 'jewish culture'. Either someone is Jewish by religion, or they are not.

If a person is born to a Jewish mother, then that person remains Jewish *unless* they join another religion. At that point, they are an Apostate, an EX Jew, and cannot call themselves Jewish any longer, though of course they are still treated with the same respect and friendship as previously.

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The word Jew (Hebrew: יהודי) is used in a wide number of ways, but generally refers to a follower of the Jewish faith, a child of a Jewish mother, or someone of Jewish descent with a connection to Jewish culture or ethnicity and often a combination of these attributes. This article discusses the term as describing an ethnic group; for a consideration of the religion, please refer to Judaism.

Jewish is a recent twist to the word Jew. It came into being after WW2 by the Jews in hopes of adding some kind of decency to an otherwise indecent people.