Judaism is a religion, but not a race. The Jews are of a single nationality, because they are one people. So Judaism is also a nationality. Why is this so difficult to understand?

Nationality is something of a 19th century (I should probably include the late 18th century in North America and parts of Europe, too) social construct. It so happens that there is identity between the Jewish people and the Jewish nationality.

The identity between a religion and a people was a commonplace in the ancient world. Each people had its religion (although of course neighbouring peoples adopted ideas from their neighbours' religions). Think of the ancient Egyptians as a people; they had their particular religion, which was found nowhere else in the ancient world. The spread of the proselytising religions (Christianity, and later Islam) changed this is the West and the Near East.

Judaism is one of the few survivors of the ancient world. Anyone religiously a Jew is therefore a member of the Jewish people. The modern era has seen secular Jews (I am a secular Jew), who are members of the Jewish people but are not religious and may not believe in God.

iain is right that there is no Jewish race. Modern antisemitism claims there is, but it's evident to anyone who's seen (say) a European Jew, a Yemeni Jew, a North African Jew and an Ethiopian Jew (ignoring the fact that each of these concepts is itself a convenient fiction) that they have no racial grouping closer than "human". But ethnic grouping is not about race. Is the "French race" really different from the "Belgian race"? What about the "Flemish Belgian" and the "Valloun Belgian" races?

None of which has any bearing on nationality, religion, or people. Each is a separate concept; they interact in various intriguing ways. Just ask an anthropologist.