The rate at which a substance conducts heat?

Thermal energy flows through any material where there are regions of different temperature. Obviously, heat flows from hot to cold.

The rate at which heat flows through a material is determined by the thermal conductivity of the material. All types of matter have some thermal conductivity. (Thermal resistivity is a related concept but is basically the inverse of thermal conductivity.)

Thermal conductivity is a materials specific property. There are tables of values available for all common substances. For all solids, the rate of heat flow is directly proportional to temperature difference. In a simple geometry, such as a flat plate with two surfaces at different temperatures, there is a simple formula to give the rate of heat flow from the hot surface to the cold one.

For fluids and gasses, the rate which thermal energy is transferred become complex because of convection processes where movement of the the liquid or gas itself occurs. This is a very geometry specific situation and there is no simple formula for calculating heat transfer rates. (People have developed useful formulas for some special cases and these are applied as needed in engineering.)

Though not immediately relevant, energy transfer can occur through radiative processes. It is necessary to mention that for completeness.

The basic answer is that a substance transfers thermal energy at a rate determined by its thermal conductivity.