For non-metals, conduction normally works through the atoms with more kinetic energy in the areas of high temperature to collide with atoms further away from the initial area. This transfers the heat to other areas of the substance. As the atoms can only vibrate about fixed positions, this is a very slow method of heat transfer.
Conduction works best in metals, due to the fact that it can transfer heat energy and electrical energy with their 'free' electrons as well as their atoms. These electrons are able to pass on heat and electrical current considerably faster due to the fact that they can move freely throughout the whole substance.
A metal at the same temperature as a non-metal (with the same mass) feels colder. This is because the metal conducts heat from your hand faster than the non-metal does.
Conduction works best in solids, and worst in gases. This is because the atoms in a solid are much closer together than those in gases. Meaning heat can be passed along faster, with less loss of energy. The worst conductor is a vacuum. A poor conductor is air, hence its use in double-glazed windows.
Bad conductors work as insulators. They can store heat for longer than good conductors, which are poor insulators, also see Specific Heat Capacity. This is chiefly why your wear woolen coats as opposed to metal ones.