Is ''a Jew'' the correct term or is it ''a Jewish person''?

The majority of Jews would look at you strangely if you were to called someone a "Jewish person" as that's not a term used by Jews.

The acceptable uses are:

He is a Jew. (noun)

He is Jewish. (adjective)

Although some people use the Word 'Jew' as a slur, that is a misuse of the Word, no different than any descriptive Word can be misused. Almost any nationality and/or ethnicity can be said as a slur by those who are bigoted. A lot depends on the context. For instance, it's probably more polite to ask someone: 'Are you Jewish?' as opposed to 'Are you a Jew?'

But it would be fine to say, for instance: Woody Allen is a Jew. Natalie Portman is a Jew.

So you can use either. Jews tend to know when someone is using the Word 'Jew' negatively, so you don't need to worry about causing offense, if you're just asking a genuine question or making a perfectly innocent remark :)