The more basic one, that most Jews would agree about, is Jewish means being of Jewish ancestry. Beyond that, Jews themselves think that being Jewish means remembering the Holocaust, being ethical, working for justice and equality, and for being intellectually curious.
Personally, as someone who is married to someone of Jewish ancestry, and whose friends over the past four decades have been almost exclusively Jewish, I've sometimes wondered why. My own musings, based on certain facts of behavior, is that Jews, especially the ones that I know, are highly intelligent, highly verbal, and interactive communicators. Conversation is rarely about taking turns and has always been a bit more intense and intellectual, but then again, that's me too.
In a previous comment, I wrote about the more salient aspects of Jewishness, but one that I have mused about was triggered by an article on The Matrix movie. I don't remember the actual reference, but it had to do with perception. I don't have the original article, but I imagined it had something to do with the multiplicity of perception.
My own personal take has more to do with wondering why all of my major personal attachments have been with Jews. The most obvious reason is that we share high intelligence, along with a strong verbal ability and an interactive style, but I have a few more personal ones. Maybe it is because we have a feeling of being successful outsiders. Maybe it is the strong concern with ethics, justice, and equality. Maybe it's my mitochondrial DNA line, of the same branch as 30% of Jews, although, in fact, I know that my spouse is not from that branch. Who knows?
I can see the point of being the successful outsider, even though one is an insider by intellect or ability, and one's outsider perception can lead to great insight, comedy being one venue. It is certainly possible for many of us to experience this sense of not quite fitting in, but the internet has had the opposite effect, making insiders of the fringe. Although our connectivity can bring together diverse and positive elements, allow us to find people, ideas and materials that were not available locally, it is also how misogynists, racists, and antisemites find their brethren.