Permeability is the ability of a substance to pass through some type of barrier or membrane. Barriers are permeable due to the presence of pores, or tiny holes that allow molecules pass through. Porosity is the measure of the density of pores in a substance. The more porous a barrier, the more permeable it will be in general.
When talking about permeability it is usually in relation to gas or liquid molecules passing through a barrier. There really is no direct measurement of permeability. The best way to measure permeability is to measure both sides of the barrier for a particular gas or liquid present. For example, in chemistry it is often important to protect chemicals from water vapor and oxygen in the air. So reagent bottles are sometimes capped with seals that prevent these things from entering into the bottle. The best way to measure the permeability of the seal would be to measure the oxygen and water content inside the bottle after a period of time to see how much has passed through the barrier. This would give you a sense of how permeable a membrane is to particular substances.