Electrolytes in the process of dissolving dissociate in solution into ions thus increasing the conductivity. Ionic liquids are compounds composed from ions with a low melting points, commonly < 100 C.
Electrolytes conducting by through the movement of those ions, but not conducting electrons.
Ionic compounds conduct electricity when molten (liquid) or in an aqueous solution (dissolved in water) because their ions are free to move from place to place. Ionic compounds cannot conduct electricity when solid, as their ions are held in fixed positions and cannot move.
Ionic liquids normally have conductivities below aqueous electrolyte solutions, around 10-3 S/cm.