The internal resistance of a battery cell measures the amount of current that is lost between the cell and its output. The cell itself contains a certain amount of energy, but some of that energy is lost in the form of heat as the energy travels from the cell, through the resistor, to the output of the battery. This measure is the internal resistance of the cell.
Batteries are not perfect circuits. Battery manufacturers build an internal resistor into them for a reason. This internal resistor regulates the output of the battery so that it can supply the desired amount of energy for a particular use. The measured output of a battery, be it DC load or AC conductance, is the energy of the cell minus the amount of resistance it must travel through. Energy is put into the circuit by the cell, but then some of this energy is taken out of the circuit by the internal resistor, in the form of heat, before it reaches the battery's output. This process is why batteries get hot.
Over time, the amount of internal resistance in a battery increases. This is because the resistor is made of metal which gradually becomes oxidized and heavier, slowing the current. When a battery fails, it is because it has built up enough internal resistance that it can no longer supply a useful amount of power to the output. If someone measures the voltage of the cell in a failed battery, they find that it is nearly identical to that of a new battery. The difference is in the internal resistance.