[Angelino97]

a ‘Jew’ is someone who self-identifies as such.

I've read that the Reform definition of a Jew is anyone who...

1. Has any Jewish heritage on either side of the family, and

2. Identifies as Jewish.

So if you have a paternal great-grandfather who was Jewish and you identify as Jewish, then you're Jewish.

[Netz]

The classic and formal definition of a Jew by something called “Halacha” (main tradition) is any person born to a Jewish mother. That’s it. Technically you can “feel” whatever you want but Jewish lineage is not passed down from a Jewish father like in Islam. It could very well be that there were Jews in your family tree for generations but if you’re born to a non-Jewish woman you’re not Jewish, not even “half” Jewish, no matter what the Reform movement says. Again, this is the classic definition.