My father is Jewish. My mother is not Jewish. Am I Jewish?
A Reconstructionist rabbi would tell you that if your mother or your father is Jewish, then you are Jewish. They would also say that if you were adopted and raised as a Jew, then you are Jewish regardless of whether either of your biological parents were Jewish.
According to Reform Judaism: "The Central Conference of American Rabbis declares that the child of one Jewish parent is under the presumption of Jewish descent. This presumption of the Jewish status of the offspring of any mixed marriage is to be established through appropriate and timely public and formal acts of identification with the Jewish faith and people. The performance of these mitzvot serves to commit those who participate in them, both parent and child, to Jewish life."
In contrast, however, an Orthodox or Conservative rabbi would not consider you Jewish since your mother is not Jewish. According to Judaism's law of matrilineal descent, Jewish identity is passed on via the mother only (not the father). The Talmud (Kiddushin 68b) explains that the law of matrilineal descent derived from the Torah (Deut. 7:3-4).
At the end of the 20th century, the Reform and Reconstructionist movements, partly motivated by the rise in intermarriage and the number of children with a Jewish father and a non-Jewish mother, decided the law of matrilineal descent needed to be reevaluated.