Does electron current flow through inside a battery when connected to an electrical load? Many popular science sites display and describe that current flows through and inside a battery when connected into an electrical circuit.
But what then prevents current flowing inside the battery between its two terminals + and - when the battery is not connected to a circuit (i.e. open circuit)?
I have read in manufacturer specifications that a battery has practically zero leakage current between its terminals and a dielectric resistance of many MOhms.
How is it possible all the electrical circuit current flowing through the battery and in the same time  there is no electron current possible between the two terminals of the battery?
I am confused?
 A: The current flowing in the battery when the terminals are connected to a load is an ion current, this resolves the contradiction of being able to conduct current but not electrons. The electrode on the negative terminal of the battery is oxidized and dissolves as it gives off electrons, and ions are created and go into solution in the electrolyte and on the other electrode a different ion takes on the electrons and gets deposited on the electrode (a reduction reaction). The redox reactions at the electrodes is what provides the energy the battery gives off.
To keep the electrolyte net neutral a (net) ion current must flow. How this is going on depends on cell design (e.g. the negatively charged ions may flow from one side of a semi-permeable membrane to the other, while the positively charged ion concentrations on both sides of the membrane change). There are also cell designs that don't need a membrane, e.g. the lead battery.
When the terminals are not connected to a load, the electrolyte and the electrodes are in equilibrium between the electrochemical potentials of the electrodes and the voltage on the electrodes. So no ion-current is flowing (as the reaction rates of reduction and oxidation at each electrode are equal at the constant voltage that is kept between them). The electrolyte solution and the case of the battery do not conduct electrons well, so the battery only discharges slowly on its own.
A detailed description of the processes involved can be found on wikipedia: https://en.wikipedia.org/wiki/Galvanic_cell.
(Of course, the electrolyte may indirectly conduct electrons due to ion-conduction mechanisms, e.g. a watery solution will undergo electrolysis, this is a real loss mechanism in some battery designs – e.g. leaky alkaline batteries are due to hydrogen production: https://en.wikipedia.org/wiki/Alkaline_battery#Leaks.)
