Question: How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any? It is impossible for humans to be watching every single wavelength. For example, what about wavelengths that are too big to detect on Earth?

Answer: 
If dark matter emitted very long wave lengths of electromagnetic radiation it would mean it is composed of charged particles. There is no escape from that conclusion. Somebody might propose that dark matter is some strange configuration of charged particles which acts as a very long wavelength antenna. That might be a good model, but there is a hitch with it. If it emits longer wavelength radiation it must be colder. The Wien displacement law is that $\lambda~=~c/T$, for $c$ a constant, which may be calculated, but is not relevant here. For $T~=~2.7$K the wavelength of radiation is $.1cm$. Let me assume that there does exist some very long wavelength radiation, say $100m$ as the lower bound on this. The ratio of temperatures gives that
$$
T\_{100m}~=~\frac{10^{-3}}{10^2}2.7K~=~2.7\times 10^{-5}K 
$$
So all this can be is a very cold gas, which is the hitch. This gas is much colder than the background radiation and not in equilibrium. So from some physical grounds this is not likely, and dark matter is most likely not some ordinary form of matter that interacts by EM.
Dark matter was originally called that because it was matter.... that was dark. In other words, it was inferred that there must be something out there, in galaxies and in clusters of galaxies, that had a gravitational effect, but which could not be readily detected using electromagnetic waves at any wavelength.

Now initially it was thought that this "dark matter" could just be stuff that was hard to detect and that had a large mass-to-light ratio. An example would be lost golf balls. Golf balls do not emit very much light - well, not unless you make them very hot. On the other hand they do have mass. If you have enough (cold) golf balls then it could be that all those lost golf balls explain why galaxies rotate too fast as a function of radius or why the galaxy velocity dispersion in galaxy clusters seems to high for the amount of luminous matter that can be accounted for. More seriously, the cosmic lost golf balls that have been considered were very faint stars, cold brown dwarfs, old white dwarfs or perhaps lots and lots of planets or mini-black holes.

Pretty much all of these possibilities have now been excluded (primordial black holes may still be a possibility) using, for example, transit and microlensing experiments.

At the same time, what we mean by "dark matter" has evolved to mean *non-baryonic* dark matter. This is not just matter which is faint, but matter which does not interact electromagnetically at all. Such matter is by definition, absolutely dark.

Why is such a material proposed? Well firstly there is the failure to detect enough baryonic dark matter (i.e. the lost golf balls which are just very dim), which does interact electromagnetically but is just dim. Secondly, we know how much baryonic matter there is in the universe by estimating the primordial abundances of helium and deuterium produced in the big bang. The amount falls short by a factor of about 6 of the amount of gravitating matter required to explain the dynamics of the universe, galaxies and galaxy clusters. The remaining 5/6 must be non-baryonic and since we can't have charged leptons like electrons without an equal number of positively charged baryons, then it does not interact electromagnetically either (any quantity of positrons would have annihilated long ago). It is therefore, by definition, dark at all wavelengths.