Dark matter is technically anything that doesn't radiate, but often it's meant to refer to WIMPs -- weakly interacting massive particles.
Now WIMPs are basically what it says on tin. We know they don't interact much, with themselves or anything else, because the rotation curves of the galaxies implies there are large halos. This means there's little to no friction, as otherwise the majority would fall down to the galactic cores, like normal matter does.
We assume they're a separate type of particle from anything we know, as otherwise that shouldn't be possible. We know they're massive, as otherwise we'd see them in particle accelerators... well, we'd see the missing mass at least.
Then, what are they? Well...
Probably a lot of things. The "dark matter sector" outweighs visible matter by a lot, and there's no reason that should be just a single particle.
But anyway, think of graph theory. You can map particles to nodes on a graph, and the fundamental forces to edges.
Not all particles are connected by way of all forces. Electrons lack color charge; neutrinos lack basically everything. (Even when there's a connection, connections can be of varying strengths.)
Dark matter is matter that lacks any connection to the other particles, except gravity.
(Could there be particles that are unconnected by gravity? That gets speculative, but probably not. Gravity is special.)
Ah, but-- there might be more than one dark matter particle, and forces connecting them that won't touch our type. For the most part we know this isn't true, since we know they don't interact with themselves either, but this is only to say that most dark matter can't be doing that.
There absolutely could be dark-matter stars, or stranger things.
... This is purely speculative, though. Good for fiction, but the only kind of dark matter we know exists is the WIMPs.