Probably the first place we meet it is in the kitchen, oven doors can get very warm on the outside even if the door is well insulated and never even think of stirring the contents of a saucepan with a metal spoon. We tend to think that conduction only happens in solid materials but this is not true, conduction takes place in solids, liquids and gases, usually though the other forms of heat transfer mask it in all but solid matter.
Heat transfer by conduction relies on direct contact between atoms or molecules. The precise method depends on the state of the matter (solid, liquid or gas) and the material involved.
This is the classic form of conduction that tends to be taught in school and as far as it goes it is accurate.
Imagine a bar of (non-metallic) material, say wood, plastic, glass or maybe even a ceramic. As one end is heated in a Bunsen burner flame that end gets hot.
In a solid each particle is locked in place by chemical bonds. The particles can however vibrate on the spot (and in the case of molecular materials the shape of the molecules can vibrate).
Example Vibrational Modes in an Atomic and a Molecular Solid
As one end is heated the amplitude of the random vibrations of the particles there increases. However the bonds allow energy to pass along the chain to the unheated end. The result is that heat is conducted through the non-metallic solid.
The easiest way to imagine this is to think of the atoms of the solid as being connected by a series of springs or elastic bands. As one end of the chain is shaken the vibration is passed throughout the whole of the solid.
Conduction in a Non-Metallic Solid
One thing to remember about non-metallic solids is that their thermal conductivity is usually quite poor. In fact if the heated end of a bar is hot enough it is likely to burn, decompose or melt before significant amounts of heat have been passed to the cool end. If this were not the case then lighting a match would always be a painful process.
Liquids are also tightly packed materials and so the particles are not really free to move about. They can however rotate or tumble, as well as vibrate on the spot like solids.
Example Rotational Modes in a Molecular Liquid
The result is that the particles are constantly colliding with their closest neighbours and this causes the energy of the particles to spread from the hot, energetic end to cool, relatively inactive end. This should mean that a liquid is as good a conductor as a non-metallic solid. However gravity usually gets in the way and a far more efficient mechanism, convection, takes over. Because of this we rarely get to see liquids conducting heat. If we can stop the liquid from flowing then conduction replaces convection.
An example of this is in a wet suit. A thin layer of water fills the fleecy lining of the wet suit but once there is unable to flow out through the rubber skin. Water is a poor conductor of heat and so the result is that you stay warm even though you are surrounded by cold water. You need to be careful though if you keep the wet suit on after you get out of the water and it is a hot day. The wet suit stops your body from losing heat by convection as it normally would and heat exhaustion and even the far more serious heat stroke can result.
Conduction in a Liquid (with convection suppressed)
click image to view animation