What does thermal conductivity actually measure? Forgive my layman, non-physicist terminology used here. Hopefully I'm not too much of a caveman to express myself properly.
What does thermal conductivity actually express? Is it measuring the amount of heat that transfers through a material? Or the speed at which the heat transfers? Or some combination of the two? Or something else?
For example, if I have a wall with such-and-such thermal conductivity and a heat source on one side, what does thermal conductivity actually tell me for the amount of heat that will be transferred to the other side, how long that will take, and so on?
Edit: if thermal conductivity is the speed of heat transfer, what am I to make of the fact that dense materials like concrete and compressed earth blocks have a high thermal conductivity (≈1.5) relative to dedicated insulation materials (≈0.04), yet heat transfers through them slowly--this property being explicitly utilized in certain applications, in fact, such as passive solar design.
 A: Thermal conductivity measures the speed at which heat energy travels through material.
That's different to the speed at which changes in temperature travel through material, which is driven by a combination of thermal conductivity and thermal mass.
So, to use your example, concrete has a high thermal conductivity: it will lose heat energy quite quickly, so a hot thing inside a concrete box can cool down quite quickly. However, concrete has high thermal mass: it takes a lot of energy to raise its temperature by 1 Kelvin. So even with heat going into it quickly, its temperature will rise slowly.
That's why concrete and earth walls are used in some passive solar designs: not necessarily for their insulation properties, but for their properties as a heat buffer: they can absorb a lot of heat for relatively low changes in their own temperature, and radiate it back out again. That gives you a wall surface with a fairly steady radiant temperature, which feels a lot more comfortable than a surface with a highly variable radiant temperature; and it gives you a huge buffer that allows you to store solar energy in the day and release it at night, thus giving you cooling during the day when you need it, and heating during the night when you need it.
A: It represent the speed actually. it is defined as:
"The amount of energy that is transferred from A to B where $AB=1meter$ and difference between the temperature of point A and B is 1 kelvin, in each second."
for example the the thermal conductivity of wood is about 0.4 . it means if you have a wood with a length of 1 meter and and $\Delta(\theta)=1 degree$ (of both end of the wood) then 0.4 joules will be transferred in each second, from one end to another.
A: One of the best ways to explain concepts like this are to use labelled diagrams, such as

with a very nice explanation from the  CBFT blog page, with a nice definition:

When the temperature of one surface of a solid material is higher than another, heat will move through the material. Depending on the characteristics of the material, this conductive heat transfer may be slow or it may occur quickly. The rate of heat transfer is defined by the coefficient of thermal conductivity.

Essentially, thermal conductivity is how fast will heat from its source pass through the material - if the material is thicker, then it will take more time to conduct through.  
It's reciprocal is thermal resistance
(not a 'caveman' question at all!)
A: The reciprocal of thermal conductivity is thermal resistivity (from this source). In analogy to ohm resistance, where the resistance depends from both the current flow and the potential difference, you are right saying that it is showing "the amount of heat that transfers through a material" and "the speed at which the heat transfers". 
A: I believe your confusion is caused by mixing two different measures of conductivity. Steady state conductivity and transient conductivity.(also referred to as dynamic heat flow) 
Steady state is the easy concept and is generally what is intended when talking about conductivity. Steady temperatures are applied to each side of a material(one side warm, one side cool) and then you wait for an extended time(hours, days, weeks) for the material's internal temperatures and heat flow to settle into a constant steady state. At this point you calculate the energy flowing(usually in watts) across the material per degree of temperature difference. This can be for a total for a complete object or per unit of area and thickness for a material property. (note that you could also use steady power input and then measure the temperature difference after waiting for it to settle, just two sides of the same equation.) 
Transient conductivity is the heat flow from momentary changes in temperature or power, and this can have several forms depending on what you are engineering, flow from side A, flow to side B, or some combination. This transient effect is seen in your example of a solid concrete wall where the material has a moderate steady state conductivity but also a large heat capacity and so thermal changes on one side take substantial time to be noticed on the far side. This heat capacity mechanism will tend to make the transient conductivity seem higher than steady state when measured on the near side(side with change) and lower when measured on the far side(side without change), it will also tend to cause delay or lag in temperatures.
A: Suppose we have flat plane of some material through which heat is passing in the perpendicular direction. For example it is a layer of material placed in front of a heater, or something like that. Let the $x$ direction be perpendicular to this plane.
Let $J$ be the amount of heat energy passing through the plane, per unit area of the plane and per unit time. So in dimensional terms this $J$ is a power per unit area. It is also often called a flux. Let $dT/dx$ be the temperature gradient at the surface. Then
$$
J = - \kappa \frac{dT}{dx}
$$
where $\kappa$ is the thermal conductivity. If you are unfamiliar with this notation then for a uniform flat slab of material you can also write it as
$$
J = - \kappa \left( \frac{T_{\rm out} - T_{\rm in}}{w} \right)
= \kappa \left( \frac{T_{\rm in} - T_{\rm out}}{w} \right)
$$
where $w$ is the thickness of the slab.
Therefore thermal conductivity is the amount of heat flux per unit temperature gradient. Think of the temperature gradient as providing a slope, and the heat wants to flow down this slope. It doesn't all rush down in an avalanche because the material (with property $\kappa$) is resisting the flow a bit. But neither is the flow totally blocked by the material. The material has a propensity to allow the heat to go through. This propensity we call thermal conductivity.
