All conductors share the property that they're able to carry an electric current (a flow of electronic charge) in response to an applied difference in end-to-end voltage. Inside the material it is the electric field (the voltage difference/distance over which it is applied) that forces any ‘free’ charges to move. In most cases the vast majority of the electrons inside the material are ‘fixed’ in place — tightly held by the atom they're orbiting. However, in all conductors some of the electrons find a way to move.
There are various ways a material can conduct...
In some materials the band-gap energy is quite small compared with the average thermal energy available per particle. This means that random thermal motions cause the atoms & electrons to jiggle around, providing some electrons with enough energy to jump the gap. This produces a thermalised population of free electrons in the conduction band and an equal number of freely movable holes in the valance band. These free charge carriers (-ve electrons and +ve holes) can then move in response to an applied field and the material conducts.
Some materials have a very small band gap, or even a non-existent gap. Then the conduction and valance bands touch or overlap. This means an electron only requires a tiny amount of extra energy to reach a free energy level.
A neutrally charged atom will have the same number of electrons in orbit as it has protons in its nucleus. This number doesn't always mean that the lowest energy levels all fill up when the electrons get as close to the nucleus as they can. Solids made of these materials tend to have an energy band which is always only partly filled with electrons. This can make it very easy for electrons to move from atom to atom since they don't first have to jump to a higher band.
The materials called ‘metals’ — iron, silver, aluminium, etc — all belong to the second two types of conductor described above. The very low energy required by their electrons to move explains why metals generally conduct electricity very easily.