With this simple experiment you can verify that the distilled water presents a very low conductivity because the potential of the first circuit is much less than the developed by the battery. The small potential of this circuit is due to the presence of the H⁺ and OH^- ions produced when some molecules of water are dissociated. However, the number of molecules of water that are dissociated is practically negligible, making the distilled water a weak electrons conductor. It is worth noting, that as the distance among electrodes increases, the resistance to the electrons transfer in the distilled water increases considerably. This fact evidences even more the low conductivity of the distilled water.

When salt is added (NaCl) to the system, the conductivity of the system increases considerably. The water not only dissolves the salt, as well as it dissociates the salt molecule. The elementary sodium (Na) presents an excess electron and the chloride (Cl) has a strong affinity for electrons. Consequently, after dissociation, the sodium loses an electron to the chloride, forming Cl^- and Na⁺ ions. The presence of these ions increases the conductivity of the water considerably. The positive ions (Na⁺) migrate to the negative electrode (linked to the pole - of the battery) and the negatives (Cl^-) migrate to the positive electrodes (linked to the pole + of the pile). The conductivity of the aqueous solution water/salt is proportional to the ions concentration in the solution.
When you try to measure the conductivity of tap water you can verify that it has higher conductivity in comparison with distilled water because this contains ions (less pure). The tap water has higher conductivity due to the presence of the ions: Ca²⁺ and Mg²⁺. And don´t forget, try the science at home!