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.
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Copper is a prime material that can conduct electricity. It is why they used to use copper cabling in networks.
insulators
insulators semi-conductors and conductors are classified by their valence band gap
Wood is an insulator.
Plastic wrapped around the wire is insulator. Not a conductor. That is why it is wrapped around the conductor wire.
Cotton wool is not a conductor or an insulator it is a semi conductor. It is sort of in the middle, electrons can travel through cotton wool but not as good as silver, bronze, metal and so on.
The poorest insulator would be the best conductor, which is one of the noble metals. Bakelite is a fairly good insulator, not too long ago electrical fittings were made from it.
Plastic is a good insulator and a bad conductor of electricity, which makes it a good material to protect people from electric shocks when handling appliances. On top of that, plastics are inexpensive and easy to shape.