OK, here is the scoop: Jews are related both by genealogy and religion, and both are important. The word "race" is not appropriate here, as there are a number of human races in the world, and they do not include the Jews.
The original Jewish forefathers were not of any particular "race" -- they were different from the other people of the land simply because they believed in a single God.
Later, however, Jews have become essentially a multi-million-strong extended family, as intermarriage with other tribes was forbidden. As centuries passed, the genealogical family has been greatly diluted by many converts on one hand and intermarriage and assimilation on the other. This is why there are black Jews, Asian-looking Jews, etc. You could be a convert without a trace of Jewish blood but still be considered Jewish.
The question of who is a Jew, and whether a person who is 1/2 or 1/4 Jewish can be considered a Jew for religious purposes, is a separate issue from OP's question. Typically, the person's mother must have been Jewish, or the person undergo a specified conversion procedure.
Hope this helps.
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