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.