Is dark matter repulsive to dark matter? Why? I think I saw in a video that if dark matter wasn't repulsive to dark matter, it would have formed dense massive objects or even black holes which we should have detected.
So, could dark matter be repulsive to dark matter? If so, what are the reasons? Could it be like the opposite pole of gravity that attracts ordinary matter and repulses dark matter?
 A: Lubos Motl's answer is exactly right. Dark matter has "ordinary" gravitational properties: it attracts other matter, and it attracts itself (i.e., each dark matter particle attracts each other one, as you'd expect).
But it's true that dark matter doesn't seem to have collapsed into very dense structures -- that is, things like stars and planets. Dark matter does cluster, collapsing gravitationally into clumps, but those clumps are much larger and more diffuse than the clumps of ordinary matter we're so familiar with. Why not?
The answer seems to be that dark matter has few ways to dissipate energy. Imagine that you have a diffuse cloud of stuff that starts to collapse under its own weight. If there's no way for it to dissipate its energy, it can't form a stable, dense structure. All the particles will fall in towards the center, but then they'll have so much kinetic energy that they'll pop right back out again. In order to collapse to a dense structure, things need the ability to "cool."
Ordinary atomic matter has various ways of dissipating energy and cooling, such as emitting radiation, which allow it to collapse and not rebound. As far as we can tell, dark matter is weakly interacting: it doesn't emit or absorb radiation, and collisions between dark matter particles are rare. Since it's hard for it to cool, it doesn't form these structures.
A: Dark matter surely has to carry a positive mass, and by the equivalence principle, all positive masses have to exert attractive gravity on other masses.
Also, from the viewpoint of phenomenological cosmology, we obviously want dark matter to attract itself. It has to attract visible matter because this is why dark matter was introduced in the first place: it helps to keep the stars in a galaxy even though they're orbiting more quickly than one would expect from the distribution of visible mass in the galaxy.
For this reason, the force between dark matter and ordinary is surely attractive. The force between dark matter and dark matter has to be attractive, too. In fact, dark matter has played the dominant role in the structure formation - the creation of the initial non-uniformities that ultimately became galaxies, clusters of galaxies, and so on. The dark matter halos are larger than the visible parts of the galaxies: the visible stars arose as "cherries on the pie" near the centers of the dark matter halos.
There's no doubt that the gravitational force between any pair of particle-like entities is attractive. This is linked to the positive mass i.e. positive energy - which is needed for stability of the vacuum (if there existed negative-energy states, the vacuum would decay into them spontaneously which would be catastrophic and is not happening) - and the basic properties of general relativity. In particular, there's a lot of confusion among the laymen whether antimatter has an attractive gravity. Yes, of course, the matter-antimatter and antimatter-antimatter gravitational forces are known to be attractive, too.
The non-gravitational forces between dark matter are almost certainly short-range forces. In particular, dark matter doesn't interact with electromagnetism, the only long-range non-gravitational force (mediated by a massless photon) we know - that's why it's dark (emits no light).
The only repulsive force that arises in similar cosmological discussions is one due to dark energy - or the cosmological constant, to be more specific. Dark energy is something very different than dark matter. This force makes the expansion of the Universe accelerate and it is due to the negative pressure of dark energy which may be argued to cause this "repulsive gravity". However, dark energy is not composed of any particles. It's just a number uniformly attached to every volume of space.
A: There is a common misconception among lay people that the phrase "dark matter" refers to actual matter. It doesn't. It simply refers to whatever is causing constant angular velocity in stars orbiting galaxies (specifically spiral arm galaxies). In short, there is absolutely no requirement that dark matter is actual matter.
