What is the meaning of "cold" in cold dark matter?
It is a description of the velocity distribution of the dark matter at very early times, before any complicated structures like galaxies have formed. The idea is that at these times, at each point in space, there are dark matter particles with a spread of velocities. For cold dark matter, that spread is very narrow.
This is important because it affects the formation of structure later on. In the Universe's initial conditions, there are minute variations in the density, at a level of about one part in 10-100 thousand ($10^{-4}$ to $10^{-5}$). Over time, these density variations are amplified by gravity, because regions of excess density tend to pull in material from their surroundings. These initial variations appear to exist at all scales. At large scales, they seed galaxies and galaxy clusters.
If the dark matter is not cold, then the velocities of the dark matter particles tend to blur out small-scale density variations over time. These variations cannot persist on scales smaller than the distance that particles randomly drift. This could lead to consequences like a reduced abundance of small dwarf galaxies.
How does this relate to a "vanishing equation of state"?
This is related to the description of dark matter as a perfect fluid with pressure $p=w\rho$, where $\rho$ is the energy density and $w$ is the equation of state parameter. The expression "$p=w\rho$" is the equation of state. For ultrarelativistic particles (with near-light-speed random motions), $w=1/3$ in the perfect fluid approximation. With no random motion, $w=0$, so the equation of state vanishes.
Note that a fluid is, by definition, a system in which particles cannot pass other particles. In a fluid, all particles at a given point move coherently; there is no spread in velocities. This is fundamentally an inappropriate description of dark matter, since dark matter particles can freely pass other particles. The perfect fluid description can work for dark matter at very large scales, e.g. when modeling the expansion of the Universe. However, it is not a good description at any scale for which the distinction between cold and non-cold dark matter is relevant.