I'm not asking a question like "Is the wood conductive?". No. I'm asking what properties do they have to have to be good conductors. Theoretically I mean. Thanks.
To be a good conductor, they need a pathway for electrons to flow. And the electrons must be as weakly bound to the nucleus as possible.
Most metals have their valence electrons delocalised in a "sea". These sea electrons are weakly bound, and can easily move from here to there. An outside electron can easily be injected here, and mingle with the others, starting the current chain-reaction. This makes all metals good conductors. Out of these, mercury is the worst conductor (has to do with the sea being ineffective--not sure;will look it up and see)
Graphite is a goodish conductor since the unpaired electrons in the sheet structure are delocalised. An outside electron can again join the delocalised cloud, and start the current chain reaction--along the sheet, though. Current flow perpendicular to the sheets is harder.
Semiconductors do not exactly have an electron sea--all the electrons are used up in forming weakish bonds. These electrons can be displaced, causing current. We can also dope the semiconductor--this changes the electron density and interesting stuff happens.
Electrolytic solutions have ions. Instead of electrons moving, these ions move, causing current.
Insulators/bad conductors have all of their electrons bound in the nucleus or used in forming strong bonds.
I think that covers all the cases...