There's a theory that "dark matter" isn't matter at all, but some modification to the currently accepted laws of physics. Wikipedia lists some of these ideas:

https://en.wikipedia.org/wiki/Dark_matter#Alternative_theori...

Ultimately, it's generally assumed to be some sort of actual matter because that's what best fits the observations. No alternative explanation has been raised that better fits the observations than the explanation that it's some unknown form of matter that doesn't interact with electromagnetic radiation very much, but still participates in gravity.

As for what those observations are, galaxies are observed to rotate much differently than if they only had the mass that was visible, galaxies cluster as if there was more mass than can be seen, gravitational lensing indicates that galaxies mass more than what's observed with light, the cosmic microwave background spectrum is different from what you'd expect from only the stuff we can see, and a fair bit more mentioned on that Wikipedia page.

Keep in mind that any alternative explanation needs to not only explain the fact that these things are different, but also the exact quantity by which they are different. So far, "dark matter" is the best explanation anyone's come up with.