Any history of the battery would be incomplete without mention of Alessandro Volta. In 1800, Volta built the first battery which was called the voltaic pile due to its stacked arrangement. The battery was a stack of alternating zinc and copper plates separated by saltwater electrolyte soaked cardboard.
This arrangement of two dissimilar materials for anode and cathode separated by an ionically conductive material immersed in electrolyte is still the standard formula for a battery. In the voltaic pile, the zinc functions as the anode, donating electrons to the electrolyte, and the copper then takes up the electrons and releases copper ions into solution. This is where the separator comes into play. The electrolyte functions as a pathway for these ions, and without the separator the electrons will move through the electrolyte as well. The separator makes sure the electrons have to go around to get from the zinc to the copper (say through a light bulb) while the copper ions can get transported through the electrolyte and separator depositing on the zinc. The separator is what allows a battery to be a useful device, and separator failure is blamed for many of the laptop fires that were in the news not so long ago.
This is all fine and good, except for that part I mentioned about the copper depositing on the zinc. Once there is no more zinc surface left, the reaction driving the electrical force no longer is present. This means the battery is "dead" as happens in a AA or any other non-rechargeable battery. In a rechargeable battery the reaction can be run in reverse by applying electricity making the battery usable again. Unfortunately, not all materials can do this and it would be some time before a rechargeable battery was discovered.