Conductivity is the ability of a substance to carry an electrical current. This current is the result of electrically-charged particles gaining energy and subsequently travelling through the material, carrying them to such a place as some of their energy can be lost, usually with the intention of powering a useful process. To explain this in more detail, I shall use the classic model of a conductive material--metal:

The conductivity of metals is a direct result of a unique type of bond. This metallic bond consists approximately of the metal nuclei and most of their associated electrons being held in a lattice structure. Holding them in place, however, are the valence electrons of each metal atom--the electrons which take part in chemical reactions.

The important property of this type of bond is that despite the valence orbitals having been expanded away from their associated atom to become a large, 
gaseous collection of molecular orbitals, the electrons still retain a strong electrostatic attraction to the metal ions in the lattice. This is due to their negative charge and the positive charge on the ions--these two charges will attract each other in an effort to minimise charge separation and hence energy of the system.

In order to carry a current, however, the electrons must be free to move throughout the metal, and indeed they are. This is because they are no longer strongly associated with any particular atom and thus can move throughout the lattice of ions under the influence of a potential difference, yet requiring only a very small amount of energy to do this.

It is important to point out the difference at this point between the metallic bond and the covalent bond, in which the valence orbitals overlap in order to allow atoms to share electrons. In the metallic bond, these orbitals are poorly-defined, having been effectively replaced by the delocalised cloud of electrons. This allows for the possibility of the metal being used as a conduit for energised electrons (e.g. in an electrical cable), since there is no inherent reason why the metal would ‘prefer’ its original electrons to those from an adjacent atom. The covalent bond, however--though expanded over perhaps several atoms--is very much stronger, forcing the electrons to stay in place.

Metallic Bond Diagram
(Figure 1: The Metallic Bond)
Image created by the author for this website