What attributes cause a substance to be nonconductive or conductive
Conductivity requires mobile electrons. Metals tend to have loosely held valence electrons so metals are generally conductive. Metals have loosely held valence electrons for two reasons: metals tend not to be very electronegative (able to stabilize electron density) and metals tend to be somewhat big atoms by nature of many metals having a d-group of electrons. D electrons are further from the nucleus on a time average basis than s electrons or p electrons. 

However, conductors do not have to be metals. Consider graphite. Graphite is $\ce{sp^2}$ hybridized. Graphite is made solely of carbon atoms - and carbon isn't a metal! The mobile pi electrons in graphite due to resonance allows graphite to be a good conductor. 

I should also note that conductivity does not always imply mobile electrons; solutions can be conductive too and this is due to mobile electrolytes (ions). For example, you will find that deionized water is completely unconductive. However, regular tap water is generally conductive due to the presence of ions (e.g. chloride ions from chlorination). Gatorade is also very conductive because it is especially formulated to contain a lot of electrolytes.