Are Jews a race or a religion?
In the Bible, the concept of "race" does not exist. The Bible talks about nationalities, but not "race." See How did the different races come about? for more details.
The word "Jew" is a derivation of "Judah." When the kingdom of Israel divided, the ten northern tribes continued to use the name "Israel." The southern split was basically made up of Judah, and so was called the kingdom of Judah. The Assyrians wiped out the northern kingdom. The Babylonians captured the southern kingdom about 200 years later, but after a 70 year captivity they were allowed to return home. Thus, the name "Judah" remained and by Jesus' time the people were called "Judeans" or "Jews" for short.
Being Jewish was both a nationality and a religious description because a person became an Israelite (Jew) by birth. At eight-days old, children became a part of the Old Testament covenant through circumcision of the boys. One could also covert to Judaism, which would make a person both a Jew in religion and a part of the nation.
During the Roman empire, the Jews were dispersed and so while they still practiced their religion and considered themselves of the nation of Israel, they were also citizens of the countries were they resided. Thus Paul was a Jew by birth and nationality, but he was also a Roman citizen -- he held dual citizenship.
Thank you. This will help with the conversation of the person in which I was having on yesterday. I had a problem trying to get them to understand the differences, but this will be big help. I am also a preacher in the Lord's Church; yes, the Church of Christ.