What are the principles involved in the determination of thermal conductivity of good and bad conductors?

The heat in a material can be transported either by electrons or the atomic vibrations. Stiffer the springs that join the atoms, less easy for the heat to transport. The stiffness of the bond in turn also depends on the electronic structure and overlap. Therefore I would say that the arrangement of the atoms, and their electronic structures determine the thermal conductivity. Generally a bad conductor of electricity (insulator) is also a poor conductor of heat (say a wood). And a good conductor of electricity is also a good conductor of heat (say any metal).