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Conductor like metals - aluminium copper steel etc.  allow flow of electricity through them. electrons move along the conductor .  this is current.

In non conductors, charge of protons or electrons is developed when additional energy is given to the atoms by some means.  +ve and -ve charge separate dueto gaining energy. When glass is rubbed due to the rubbing the surface of glass develops charge. similar behaviour on leather surfaces etc. static charge is als o developed when another charged body comes near its surface.
THe static electricity - is just electric charge - remains there on the body. we can hear some sounds, feel them. the static charge gets discharged when we touch that material with wet cloth or wet hands..

Electricity implies that the electrons (or holes) are moving and static electricity is just a build up of static (non-moving) charge.