Jews are an ethnic group, though Judaism is also a religion. People can be ethnically Jewish though irreligious -- many Jews are.
[This has been the effective definition of "Jew" in many past incidents of anti-Semitism. M]any anti-Semites hate Jews without regard to their religion; the Nazis went after the irreligious Jews as well as the devout Jews, and so did the Soviets. Much anti-Semitic propaganda focuses on Jews' supposed ethnic or cultural traits, not their religion. Nor is this just an anti-Semitic view; as I understand mainstream Jewish religious teachings, someone whose mother is Jewish, which is to say generally someone who is ethnically Jewish, is "Jewish" for purposes of Judaism even if he is an atheist.
I realize that there's some fuzziness about the definition of "ethnicity" (it usually turns on people's descent, but descendants of converts to Judaism may often be treated as ethnically Jewish, just as descendants of people who moved to Ireland not long ago may often be treated as ethnically Irish -- especially when the descendants are now not in Ireland in any more, and especially if they characterize themselves as Irish). [But s]uffice it to say that an ethnic group is a group that's usually linked by descent and culture, and that perceived itself and is perceived by others as an ethnic group. We need not delve further into this here, except to say that Jews are often treated as an ethnic group much as are Irish, Poles, Gypsies, and so on.
I prefer to use the term "ethnicity" rather than "race" to refer to Jews. Historically, however, the term "race" has also included what we now think of as ethnicity, so Jews, Italians, Irish, and such were sometimes called "races" rather than just ethnic groups. [Likewise, some old U.S. statutes and some more recent foreign statutes have been understood as using "race" to include "ethnicity."]
Jews: an Ethnic Group and Not Just a Religious One: