As noted previously, thermal conductivity is a thermodynamic property of a material.  From the  State Postulate given in thermodynamics, it may be recalled that thermodynamic properties of pure  substances  are functions  of two independent  thermodynamic  intensive  properties, say temperature  and pressure. Thermal conductivity of real  gases  is largely independent  of pressure  and may be  considered a  function of temperature  alone. For solids  and liquids,  properties  are largely independent of pressure and depend on temperature alone.

Let us try to gain an insight into the basic concept of thermal conductivity for various materials.  The  fundamental concept  comes  from the  molecular or atomic  scale  activities. Molecules/atoms  of various materials gain energy through different mechanisms.  Gases, in which molecules are free to move with a mean free  path sufficiently large compared to their diameters,  possess energy in the  form of kinetic energy of the molecules. Energy is gained or lost through collisions/interactions of gas molecules. 

Solids,  on the  other hand,  have  atoms/molecules which are more  closely packed which cannot  move as freely as  in gases. Hence, they cannot  effectively transfer energy through these  same mechanisms.  Instead, solids may exhibit energy through vibration or rotation of the nucleus. Hence,  the energy transfer is typically through lattice vibrations.  Another important mechanism in which materials maintain energy is by shifting electrons into higher orbital rings.  In the case  of electrical conductors  the  electrons  are  weakly bonded to the  molecule  and can  drift  from one  molecule  to another,  transporting their energy in the process.  Hence, flow  of electrons, which is commonly observed in metals,  is an effective  transport  mechanism,  resulting in a  correlation that  materials which are  excellent  electrical  conductors  are  usually excellent thermal conductors