Why are materials that are better at conducting electricity also proportionately better at conducting heat? It seems like among the electrical conductors there's a relationship between the ability to conduct heat as well as electricity. Eg: Copper is better than aluminum at conducting both electricity and heat, and silver is better yet at both. Is the reason for this known? Are there materials that are good at conducting electricity, but lousy at conducting heat?
 A: This is true only for metals. Diamond, for example, is barely a semiconductor. But it has a better heat conductivity than any metal.
A: What links the two conductivities is that they both depend on how transparent the material is to electrons traveling around the Fermi energy.
Thermal conductivity also has a contribution from lattice vibrations, but for metals the contribution from electrons dominate.
In an analogy, imagine two recevoirs of water connected by a channel. The two recevoirs and their height correspond to the leads of the conductor at different potentials and the width for the channel at the surface corresponds to the transmission of electrons at the Fermi surface. 
Changing the height of the water in one recevoir in relation to the other will create a net flow of water that depends linearly on the width of the channel. This is the analogy of the flow of electrons through the conductor.
Temperature in this picture correspond to the amount of waves in the recevoirs. And how fast wave energy in one recevoir moves to the other recevoir also strongly depend on the width of the channel.
A: See http://en.wikipedia.org/wiki/Thermal_conductivity   In metals, I think it generally has to do with the higher valence band electron mobility, but it gets more interesting elsewhere.

In metals, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann-Franz law, as freely moving valence electrons  transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator.

