If a Jewish woman lives with a non-Jewish man and has a child from him, the child is considered Jewish. When it comes to a person’s status as a Jew or a gentile, the determining factor is the mother’s status. Hence, the child of a Jewish woman is a full-fledged Jew, even if the father is not Jewish.
If a daughter was born from a relationship between a Jewish woman and a non-Jewish man, is she permitted to marry a Kohen?
Although instinctively we might assume that she may not marry a Kohen, in truth, such a girl does not fall under any of the categories of women forbidden to a Kohen. The Torah forbids a Kohen from marrying a divorcee, a "Halala" and a "Zona." A "Halala" is a woman born from the union between a Kohen and a woman forbidden for Kohanim (or the daughter of a "Halal"), so a girl born from a Jewish woman and non-Jewish man certainly is not a "Halala." And a "Zona" is a woman who had engaged in a forbidden relationship. In the case under discussion, the mother had engaged in a forbidden relationship, so she would certainly be considered a "Zona," but the daughter has not done anything to fall under this category.
This issue is actually subject to a complex debate among the Rishonim (Medieval Halachic scholars). The Gemara makes mention of a Rabbinic enactment forbidding the girl in such a case from marrying a Kohen, but the Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204) does not include this prohibition in his Halachic code. In his view, it is permissible for the daughter of a Jewish woman and non-Jewish man to marry a Kohen. By contrast, the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327) and other Rishonim accept this prohibition as normative Halacha. A third view is that of the Rif (Rabbi Yishak of Fez, Morocco, 1013-1103) and the Ramban (Rabbi Moshe Nahmanides, Spain, 1194-1270), who held that the woman in this case has the status of "Safek Peguma," meaning, it is uncertain whether or not she is Halachically eligible to marry a Kohen.
The Shulhan Aruch (Eben Ha’ezer 4) follows the view of the Rosh, that the daughter of a Jewish woman and a non-Jewish man may not marry a Kohen. However, the Halachic authorities write that if such a woman did, in fact, marry a Kohen, they may remain married, and do not need to divorce.
It goes without saying that a competent Halachic authority must be consulted for guidance concerning such a situation. More generally, it behooves all people to determine the lineage of a prospective spouse before marriage, and to consult with a Rabbi if there any other uncertainties. This is especially true when dealing with Kohanim, regarding whom the laws of marriage are particularly complex.
Summary: The daughter of a Jewish woman and non-Jewish man is considered Jewish, but, generally speaking, should not marry a Kohen. Nevertheless, a competent Halachic authority must be consulted for guidance in such a situation.
