Actually, dark matter may not be ordinary matter, but completely unrelated and unknown. Ordinary matter feels the effects of gravity, the electromagnetic force, and the strong and weak nuclear forces. Antimatter feels these same forces and is very much like ordinary matter, but with electrical charges (and some other properties) reversed.
In some theories, dark matter feels only the gravitational force, and could pass right through us undetected because it does not possess electromagnetic properties at all. In other theories, dark matter is, in fact, just rather cold ordinary matter.
By the way, neutrinos probably do have a tiny amount of mass.