Being half-Jewish is a funny thing. It's not something that's easily defined. OK, so you have a Jewish parent and non-Jewish parent. What could be simpler?
There are a lot of half-Jews out there. Some have speculated that, in America, there are even more half-Jews than "full-blooded" Jews. There are half-Jewish celebrities: Bill Maher, Carly Simon, Marcel Proust, Gwyneth Paltrow, etc. I think there's a book about being half-Jewish too.
Orthodox Jews generally don't believe in half-Jewish. Jewishness, according to the orthodox practitioner, is matrilineal, i.e. inherited only through the mother. If your mother is Jewish, you're Jewish. You can denounce God, believe in Jesus, deny your family heritage, whatever. It doesn't matter. You're still Jewish. If only your father is Jewish, your status as a Jew is invalidated by intermarriage. You're not Jewish. There are no fractions.
To a reformed Jew, you're Jewish if you say you are. Intermarriage is so commonplace now that some sort of protocol for conversion is necessary.
But being Jewish isn't just a religious affiliation. There are non-religious, secular Jews with a devotion to the cultural and/or political trappings of Judaism. There's stereotypes of Jewish behavior, phenotypes of Jewish appearance, archetypes of Jewish personality.
If you're half-Jewish, chances are you will have a few or many of these traits, even if you were not raised as a Jew. You might have a big nose, or bushy eyebrows. You might have a telltale Jewish name. You might be wound-up and neurotic. You might be circumcised. The list goes on. You could be any or all or none of these things.
You could be any or all or none of these things, and if you're not raised with an awareness of it, you might never know. You grow up never knowing you were ever anything other than "normal." Or, maybe you know you're not normal, but you don't know why. Maybe kids on the playground hurl insults you don't understand. Or maybe not. Maybe you just sense it, that people treat you differently in some way, that they know something about you that you're somehow unaware of yourself. It becomes a strange birthmark, a dim prophecy, and you don't know what it means. And you grow up and you learn more and you still don't know what it means.