The term "dark matter" refers to a number of hypothetical substances proposed to provide the mass that galaxies and galaxy clusters need to have to spin as rapidly as they spin and to stay in cluster formations with one another--mass which the observable parts of galaxies contain only a tiny fraction of and which by best calculations is too evenly distributed to be hidden in massive objects at given points like black holes.
There currently exists no universally acceptable explanation of dark matter--in fact, some physicists claim instead the nature of gravity is something other than expected at large scales, Tensor-vector-scalar gravity and Modified Newtonian dynamics being two such attempts to explain how observed evidence of galactic astronomy coincides with known laws of physics without evoking matter that has never been observed.
In general, physicists have preferred imagining unobserved matter over changing already-known gravity to solve this problem, inventing "dark matter." Among the hypothetical candidates for what this substance could be are numerous descriptive categories, including "hot," "warm," and "cold" (and combinations of the three). Among these, cold dark matter is generally favored, which itself has three possible explanations that have been proposed (and an unknown number of theories that no one has proposed as of yet)--axions, MACHOs, and WIMPs. Of these three, WIMPs--"Weakly Interacting Massive Particles" seems to be the most popular.
"Weakly interacting massive particles" would mean that you and I and everything we know is being passed through at every moment by a barrage of particles that are so numerous and so large that they actually outweigh normal matter by a factor of something like six to one. So roughly one seventh of all that exists would be the visible things that you and I can see--six sevenths would be this invisible stuff thought to permeate all parts of the galaxy we inhabit in a footprint roughly the same as shape as the galaxy itself.
These particles would not interact with electromagnetism, which means light--which is electromagnetic, would not detect them. They would form no covalent or ionic bonds like atoms do, so it would seem they'd build no complex arrangements of particles--nor would they react to the essentially magnetic fields of bonded atoms, making it possible for them to float right through objects without been seen or felt. (Really, this should be called "invisible" matter rather than "dark matter," but the term is what it is.) They would interact with gravity (this is the whole reason they were invented in the first place) but if they were evenly distributed over, say, the solar system, their greater-gravity-attraction than normal matter (by outnumbering normal matter) would not be detected because their distribution would pull equally from all sides at once on this scale, balancing one another out.
As a source for speculative fiction story ideas, dark matter has a number of possibilities, only a few of which I'll explore in this post. One would be to imagine that dark matter particles do bond with one another in complex arrangements, based on physical principles that have not yet been discovered. So scientist X discovers a means by which dark matter can be visualized (say with a focused neutrino beam or something like that)...and he discovers that we human beings are surrounded not only by invisible matter, but that this matter is linked together into structures of some kind. Perhaps the structures could appear to be natural and uninhabited, or perhaps they could appear that way at first, the scientist later discovering somehow that beings inhabit this invisible world of particles of differing physical reactions than we have. And that we are every bit as invisible to them as they are to us...or perhaps, they would see objects such as our sun because of its massive generation of neutrinos, but just barely...while we would be completely invisible to them.
For the story's sake, this physicist would find a way to interact with these beings of dark matter. Perhaps he or she would even find a way to enter into their kingdom, transforming himself into their substance through some sort of Star Trekish transporter-like analogue, which would be set to convert normal matter into this strange "dark" stuff.
Such beings in complex arrangements completely independent of us, based on physics human scientist have not even imagined, made of dark matter, inspires another story setting for me. What if the situation with dark matter were much more subtle? So the physicist investigating them sees them only as particles...but then later discovers that on a large scale--as in the size of a galaxy--and over lengthy time, limited by light speed and the relative low field strength of gravity--the particles interact with each other in a way that mimics the interactions of human brain cells in a strange slow way. As if the entire galaxy is shadowed by a single intelligence, a single consciousness made of dark matter. Other galaxies would also prove to be individual minds...and normal matter would be for these minds just a strange and unexplained little set of corpuscles inside their massive bodies...