Intersectionality is the idea that various forms of privilege and circumstance interact with each other to make an individual. Certain influences are more impactful upon a particular person's circumstances, and thus influence privilege to a much greater extent. The non-linear nature that DinosaurThussy is talking about can better be shown with examples.

If you're homeless and white it's clear that you're in a worse off situation than a billionaire who is black. Class status has a far greater influence on this situation. It would be fair to say that the black billionaire has more privilege due to his class status but not his ethnic identity. That being said, it's unlikely that the white man was denied a job due to his race in a way a homeless black person may be. Being poor and white and poor and black have many commonalities, but intersectional analysis allows us to understand the different ways and avenues that particular characteristics influences the ways that a person may end up in a particular circumstance.

The idea continues on. A person who is a billionaire may be significantly shielded from a lot of racism, or face it in a less extreme way. For example, that proverbial black billionaire likely wouldn't have many run ins with racist cops in impoverished neighborhoods. However, he still might face the unifying characteristic of being called a slur by his peers in the way that a poor black person might. His privilege of wealth may not complete inoculate him facing racism at all, even if he faces it in a less extreme way.

In essence, this situation is viewing individuals dialectic-ly. It seeks to understand how all of a person's identity and circumstances relate to the struggles and oppression certain groups or people may face in society.

That poor mine worker is still in a better position than an otherwise identical minority would be in the same position.

A poor mine worker is in a tough place but at least he wasn't refused that job because the company doesn't hire non white people.