Heat transfer by conduction

Mechanism
At all temperatures, heat energy (associated with temperature) is transferred by conduction between any two points at different temperatures when these points are connected by a heat transfer medium. The medium may be a solid, a liquid, a gas, or a plasma, i.e., the primary states of matter. Since macroscopic temperature is a reflection of the thermal energy of the atoms involved in the solid, the transfer medium simply provides a way that atoms/molecules tend to share their thermal energy with those around them; as a consequence, hotter regions become cooler because they dissipate some of their thermal energy to surrounding cooler regions which are in thermal contact.

The conductive heat transfer between two regions or bodies at different temperatures is proportional to the temperature difference between the regions, with the proportionality constant dependent on the physical and dimensional properties of the material between them:

Dynamics
The time-dependent behavior of the system above is determined by the fact that energy transfer (power) from the hotter to the colder body is proportional to the temperature difference between them at that time. Initially, then, the power transfer has its highest value, but as the two bodies approach the same temperature the rate of change is slower. This leads to an exponential approach to the same final temperature for the two bodies. In addition, the final temperature will depend not only on the initial temperatures of the two bodies (and in reality of the thermal link between them), but also on the masses of the bodies at the different temperatures: if the cold body is much larger than the hot one, the hot one will cool almost to that of the cold one, while if the masses are the same the final temperature will be midway between the two initial temperatures.

Generalization
This behavior actually appears much more broadly in engineering. For a description, click here.

Material properties dependence
Conductive heat transfer may occur through solid materials, or through liquids or gases or plasmas. In any case these materials must be in contact with the bodies for conductive heat transfer to take place between them. The thermal conductivity of gases or plasmas is much less than that for solids or liquids, because the separation between atoms or molecules in the gas phase is far greater than in the solid or liquid phase. For example, the thermal conductivity of aluminum is about 10,000 times larger than that of air.

In addition, the thermal conductivity of solids varies widely, dependent on the structure of the solid. For example, metals such as Al and Cu have high values (2.37 and 4.01 W/cm-K), while insulators can have much lower values (quartz [SiO2] at 0.0138) or relatively high values (Al2O3 at 0.363). The thermal conductivity of gases depends significantly on their pressure, because this determines their interatomic or intermolecular spacing.