Heat conduction, also known as thermal conduction, is one of the three modes of heat transfer - the others are convection and radiation.
Thermal energy flows spontaneously from bodies at higher temperatures to bodies at lower temperatures until the bodies are at equal temperature. Other things being equal, the greater the temperature difference, the higher the flow rate of thermal energy.
Thermal energy is related to the average kinetic energy of atomic-scale particle movements within matter - the more vigorous the movement of particles, the higher the temperature.
Conduction of thermal energy takes place when local interactions between atoms and molecules and electrons pass kinetic energy from particle to particle. (There is no net movement of particles in a flow of thermal energy or heat.)
Heat transfer in fluids is by random motion of atoms and molecules; in solids, heat is transferred by electrons and phonons.
In general, thermal conductivity is greatest in solids, then liquids, then gases.
Metals are generally good conductors of heat for the same reason they are good conductors of electricity. Their mobile electrons are excellent carriers of thermal energy.
Not all non-metals are poor conductors of heat. For example, carbon in the form of diamond is a poor electrical conductor, but is an excellent conductor of heat. The explanation lies with phonons - i.e. lattice vibrations that play a fundamental role in thermal conduction in solids. Diamond's uniform, continuous 3D network of covalently bonded atoms scatters phonons to an unusually low extent, hence phonons carry heat particularly efficiently in diamond.