In the following animation neutrally-charged copper atoms form a metallic bonding. That means they order in a solid atom structure and give away one electron that can move freely between the atoms. That means the atoms are then positively-charged. The whole metal is neutrally-charged because the freely moving electrons cancel the positive charge of the atoms.
After adding electrons to an electrical conductor, there is an excess of electrons which means the conductor is negatively-charged.
After removing electrons from an electrical conductor, there is a shortage of electrons which means the conductor is positively-charged.
When a voltage is applied to a neutrally-charged conductor, the electrons move towards the positive pole. The same number of electrons that is going out of the conductor at the positive pole is coming in at the negative pole. As such the electrical conductor stays neutrally-charged.
In the following experiment a strongly positively-charged and a weakly negatively-charged metal sphere bump against each other. Instantly charge balancing takes place that is electrons move from the negatively-charged sphere to the other sphere so that both spheres are equally charged.
In insulators the electrons cannot move freely but bound to the atoms. As such in insulators no current flow is possible.