Why do electrically conductive materials tend to be thermally conductive? It's hard finding substances that are good insulators of one and conductive of the other. Why the correlation? And why do some materials not have this correlation?
 A: Because both heat and electricity are mediated by the same particles: Electrons!
Conductors usually can be modeled by thinking of electrons as if they're free... flying inside the material. This is called the Free Electron Model. Hence they can travel from one end to the other easily.
Electricity is transferred when the electrons face an electric field.
Thermal conduction happens when the electrons gain high kinetic energy and diffuse faster to the rest of the material (ignoring the lattice, phonons (quanta of vibrations, like photons being quanta of light/electromagnetic field) and other stuff)... I'm just explaining here why there's a relation between thermal and electrical good conductors.
As to the last question, why are some materials have discrepancy between electrical and thermal conductivity: Such things are studied on case-to-case basis. If it's not a good conductor for both, then the free electron model isn't a good model for it. One then has to study the band structure of the material and see how it acts to heat and electric fields. It's always easier to explain correlations, not discrepancies. So why some materials are not like that? Because they're not good candidates of the free electron model.
