Conduction is the transfer of thermal energy without any flow of the medium. Substances are made of tiny particles called atoms or groups of atoms called molecules. When a solid is heated the molecules begin to vibrate vigorously and bump into neighbouring molecules, transferring kinetic energy, making them vibrate as well. Thus, thermal energy is transferred from the hot end to the cold end.
This method of thermal transfer through molecules is common to both metals and nonmetals. However, in metals a second process of heat transfer takes place, through the diffusion of free electrons. These electrons can move throughout the metal, and when heated the free electron moves faster as a result of a gain in kinetic energy. These fast moving electrons carrying energy diffuse into the into the cooler parts of the metal and transfer their kinetic energy through collision.
Liquids and gases are much more poor conductors compared to solids. Conduction in these two states involves the transfer of fast moving molecules to slow moving ones through collisions. These collisions are not very frequent in liquids and even less in gases where the molecules are further apart.
The two mechanisms for conduction are molecular vibration and electron diffusion.