Well, yes and no; their race is Semitic; actually the same as Arabs and other Middle Easterners.
It's a branch of a language family. "Race" doesn't really have any meaning in this context.
Aethiopians are "black", but they are Semites. Berbers are white, yet all of them live in Africa and always have. The only branch of the language family that is almost exlusively black is the Cushite branch (Somalis etc.).
The word "Jew", as I've always understood it, is kind of slang-ish. A bastardization of "Judaism", which is, of course, their faith. However, I could wrong.
Jew is the English version of Hebrew "Yehudi" which refers to an inhabitant of the former kingdom of Yehuda (Judaea) and such an inhabitant's descendants. The Kingdom Yehuda was inhabited by the Israelite tribes of Yehuda, Simeon, and Binyamin. The people of Israel consist (by Mosaic law and still recognised) of 12 tribes (and a priesthood) and the other 9 tribes lived in Samaria. One of those 9 tribes left for Egypt over 2500 years ago.
"Judaism" is the religion of the Kingdom of Judaea. The Kingdom of Samaria (the other 8 tribes) and the tribe who left (today's Ethiopian "Jews") have slightly different religions.
So the word "Jew" ultimately refers to a member of one of the Israelite tribes, technically to members of three of those tribes.
Non-Jewish Israelites still exist, including Indian "Jews", Ethiopean "Jews" and Samaritans in today's state of Israel. But 95% of Israelites are Jews (or of the priestly tribe) and hence the word "Jew" and "Israelite" can be used almost exchangeably, especially since all non-Jewish tribes except the Samaritans are today also "Jews" by conversion.
Bnei Menashe (India, descended from a northern tribe): 12,000
Jews (descended from the southern tribes): 13 million
Beta Yisrael (Ethipia, descended from tribe of Dan): 150,000
Samaritans (Israel, descended from northern tribes): 712