Interfaith marriage in Judaism has historically been looked upon with very strong disfavor by most Jewish faith leaders, and today continues to be very controversial. The Reform, Reconstructionist and Progressive Jewish movements have no firm prohibition against intermarriage, and are willing to officiate at interfaith marriages. Non-Jewish spouses are not necessarily encouraged to convert to Judaism anymore. Most rabbis from these denominations do still try to persuade intermarried couples to raise their children as Jews.
All branches of Orthodox Judaism follow the historic Jewish attitudes to intermarriage, and refuse to accept them as legitimate and forbid sexual relations with a member of a different faith. Orthodox rabbis refuse to officiate at interfaith weddings. Secular intermarriage is seen as a rejection of Judaism.
The Conservative Movement in Judaism does not sanction the Jewish validity of intermarriage, but does encourage the acceptance of the non-Jewish spouse within the family.
The Orthodox and Conservative movements only accept someone as Jewish if their mother is Jewish or if their mother has undergone a conversion to Judaism performed according to classical Jewish law. The Reform movement accepts someone as Jewish if their mother or their father is Jewish. So if a Jewish person wants to marry someone who has a Jewish father, but a non-Jewish mother the Conservative and Orthodox movements will consider that an interfaith marriage.
Thank you to Gail Katz