What is voltage, really, inside a battery?

Voltage inside a battery is the push a chemical reaction gives to electrons. A battery does not store electricity like a bucket stores water. It stores chemicals that would love to react, and it arranges things so that the reaction can only go ahead if electrons take a detour through your device. The more eager the reaction, the harder it shoves those electrons along, and that shove, measured in volts, is the battery's voltage.

The reaction in question is a redox reaction, short for reduction and oxidation. Oxidation means a substance loses electrons, and reduction means another substance gains them. The two always happen together. In a battery they are pulled apart: one electrode gives up electrons, the other takes them, and the only road between them that electrons can use runs through the wire and the light bulb or phone in the middle.

Why is a fresh alkaline cell about 1.5 volts, while a lithium cell gives 3 volts or more? Because voltage depends on how much energy the chemistry releases for each electron, not on how big the battery is. Lithium reactions release a lot per electron. A volt is exactly that: energy per electron moved, or more precisely per unit of charge. At 1.5 volts, each coulomb of charge that flows does 1.5 joules of work.

Alessandro Volta never knew any of this. In 1800 he stacked copper and zinc discs with brine-soaked paper and got a steady current, but he thought his pile was an endless source of energy. Michael Faraday showed in 1834 that it was the chemistry being used up. A feeble copy can still be made from two coins and a paper towel dipped in salt water.