The above website explains what Intersectional Feminism and Kyriarchy are. But I’ll give a general gist here, with the caveat that the website above explains in much better detail.
Kyriarchy is the complex hierarchical system in which we live. It takes into account race, gender, sex, language, class, sexual orientation, (dis)ability, culture… etc.
Intersectional Feminism is concerned, not just with patriarchy, but with kyriarchy.
I am a woman, and thus disadvantaged as compared to a man. That is patriarchy.
However, I am also white, cisgendered, heterosexual, middle-class, English-speaking in a predominantly English-speaking country, physically non-disabled, of a culture which is not discriminated against in my country, etc. In all of these areas, I am quite privileged. I cannot pretend to know what it is like to be a Middle-Eastern, LGBTQ+, lower class, non-English-speaking, Muslim, disabled person in America right now. I don’t know anyone who fits that description either, but it has to be difficult. Now place that person in a country that doesn’t have the wealth or anti-discrimination laws that the United States has. Make that person a black Christian in China. I can’t imagine how difficult and terrifying it must be for that person to get out of bed every morning.
That is kyriarchy.
That is why I am an intersectional feminist. That is why I feel that my country and my world will never truly be free until all subgroups of society are treated equally. That is why I believe in same-sex marriage and adoption and feminism and egalitarianism in general. Because white, cisgendered, heterosexual, upper class, English-speaking, non-disabled, non-Muslim men should not be the sole first-class group in the United States. They should be treated with respect, but so should I. So should the LGBTQ+ community. So should disabled people. So should non-English-speaking people. And so on. I cannot claim to know what it is like to not be able to be legally married. I was born with the right to vote. I was born with the eventual right to marry the person I love, because I was born heterosexual and cisgendered. Not everyone was. And that is extremely unfair, and just because I can never know what it’s like doesn’t mean I can’t recognize the injustice. Just as white, cisgendered, heterosexual, English-speaking, non-disabled, etc. men can never know what it is like to be disadvantaged, but can still recognize the injustice and identify with intersectional feminism. And I, for one, really appreciate it when they have the bravery to do so publicly. Because the more people in general who speak up in support of the movement, the more seriously we are likely to be taken.