Asking out of pure ignorance here, is Jewish the religion or the race? Or is it. Both?

So, first of all, there isn't such a thing Jewish race. There are Jews of all racial backgrounds and skin colors - and I'm not just talking about converts, there are people who were born Jewish and who's families have been Jewish for generations from all over the world, of many diverse racial groups.

Judaism is an ethno-religion. It's a holdover group from a time when there was no difference between ones ethnic (cultural, familial, linguistic, national, tribal) group and ones religion. More or less, that's still true for Judaism. If you are part of the ethnic group of being Jewish, Judaism is the religion that is prescribed for you (although you don't have to follow it, obviously), and no one can practice the Jewish religion who is not part of the Jewish people.

So, Judaism as a whole is like a family. Our religion is part of the culture of the family - participating in the religion of Judaism is like going to a big giant family reunion.

Anyone can distance themselves from the family they were born into - so, people born Jewish can be religiously apostates, and not practice Judaism.

There are also probably more distant cousins in your family than you realize - so too, many people who's families used to be Jewish have now assimilated into the dominant culture and left the family, though they could come back if they wanted to. This describes assimilated Jews and people of Jewish descent who aren't Jewish themselves.

Also, being born into a family isn't the only way to join one. You can join a family through marriage, or adoption. This is basically what conversion is - being adopted by the Jewish people. Converts are still racially whatever they were before, but they are now part of the Jewish nation, and the Jewish ethnicity.

So, "Jewish" is an ambiguous term. It can mean "someone who practices the religion of Judaism and is also ethnically Jewish" or it can mean "someone who is ethnically Jewish regardless of religion."

"Jew" or "Jews" is only potentially acceptable if youre specifically talking about the religion. For examples, it would be normal and fairly okay to say "Jews, Christians, and Muslims," because all of those are nominalized. But it would be weird and inappropriate to say "Jews" in an ethnic sense. You would (hopefully) never say "Jews" in an ethnic sense the same way you would hopefully never say "Blacks" in an ethnic sense, because it has been used to depersonify us. More correct is "Jewish people." (YMMV. Some people don't like "Jews" even when referring to religion, some people don't like hearing it from gentiles, some don't care at all, etc.)