All* materials conduct electricity. The question is exactly how much do they conduct. Materials such as metals, semiconductors, graphites, etc tend to conduct electricity better than others like plastic, paper, wood, etc due to loose valence electrons that allow electric current to flow through them. Good conductors will have a low electrical resistivity. The resistivity of some common resources are shown:

Gold: 22 n Ω·m

Copper: 17 n Ω·m

Aluminium: 28 n Ω·m

Silicon: 1000 Ω·m

Quartz: 10^16 Ω·m

Rubber: 10^13 Ω·m

Note *: In 2008 scientists discovered a property called a superinsulation that in some materials at a very low temperature exhibit infinitely high resistance. Superinsulation is essentially the exact opposite of superconduction. It's mechanism is not very well understood and is an active area of research in condensed matter physics.