Have asked a few questions, this is what I got.
It's been said that Judaism is "more like a nationality than like other religions," because traditionally, a person must be born Jewish or go through a lengthy conversion process. The strictest definition of a Jew is one whose mother was Jewish.
This doctrine of matrilineal descent has been part of Judaism since at least the second century C.E. (common era), when it was codified into the Talmud, the body of religious writings that supplement the Jewish holy book of the Torah. The Talmud expands on the Torah passages of Deuteronomy 7, which oppose intermarriage by Jews. This verse states that the child of a Jewish woman and a non-Jewish man will be Jewish. Other Torah verses admonish Jewish men from taking non-Jewish wives because their children will not be Jewish.
Why and how matrilineal descent evolved is unclear. Other aspects of lineage, such as one's affiliation with the Jewish tribes, are traced through the father. And in the Hebrew Bible, paternal lineage often plays a more important role. Scholars have suggested various reasons for the use of maternal lineage in determining Jewish status.
For one, a child's mother is always known, while the father couldn't be positively identified before modern technology. Also, the Jewish people suffered a long history of oppression, during which Jewish women were sometimes raped by their oppressors. Instead of casting out the woman and the child born of the rape, the Jewish community took them in by considering the child to be Jewish.
Responding to the rise of intermarriage among American Jews, the Reform movement of Judaism took the radical step of redefining who is considered Jewish. The Central Conference of American Rabbis declared that the child of one Jewish parent, mother or father, is presumed to be Jewish. However, children of mixed Jewish parentage should establish their Jewish identity through Torah study and public, formal acts such as a Bar/Bat Mitzvah. This Reform position is controversial among Jews and not typically recognized in Israel today.