Jews are rather obviously an ethnicity. They are equally obviously a race. In the first case because a pretty large number of Jews see themselves that way, in the latter, to the extent that antisemitism exists, they're a race.

Race is largely defined by others - e.g. Black americans are a race because society sees them that way.

Ethnicity is mainly a question of self-identification - e.g. African Americans are an ethnic group because they say so.

Note that it is definitely not necessary to be racist to understand a group as a race - it can be merely and acknowledgement of social reality. Thus for example, Jews aren't really a race in the US, at least in the NE (the part I'm familiar with) but they certainly are in Poland.