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.
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...
A case not covered in the answer given by Manishearth is plasma, i.e., ionized gas where charge carriers (usually electrons) can move freely except they collide with ions. At high temperatures these electron-ion collisions become infrequent which means that electrical conductivity grows with the temperature. But even at modest temperatures, plasmas are generally very good conductors. For example, at 1 eV (about 10,000 K) the electrical conductivity of hydrogen plasma is about equal to that of copper.