Visible normal matter and invisible dark matter respond to gravity in a similar way, forming clouds of stars, the galaxies. The first tip-off was that for the amount of matter seen through a telescope, the implied mass seen should have a certain rate of spin. The galaxies actually were turning much too fast, which was only possible if they had an unseen component adding extra heft.
The dark matter and normal matter interact through gravitational attraction, so a grouping of the two tend to approach an average velocity when intermixed. A missing component for dark matter is the lack of interaction with light, thus no photon acceleration of matter as with a solar sail. Based on that difference, the dark matter will always find a path which deviates somewhat from the normal matter grouping. Other properties may apply such as carrying a static charge or self repulsion? May not stick together to build large bodies; but, still has a gravitation component.
Another observation is the bending of light from distant galaxies by something invisible in the foreground. Sometimes a duplicate image of the distant galaxy is visible offset from the primary location, or a distorted ring of light is present, with the observed galaxy at the bulls-eye. Gravity warps space—bends light, therefore has an effect like an optical lens to form, or sometimes distort an image.
This will have to be my disclaimer:
As the physicist Niels Bohr famously said, “If you aren’t completely confused... you haven’t understood it.”