Dissect a battery
You can dissect a normal carbon zinc battery by cutting it open with a hacksaw.(These often say "Heavy Duty" on them.
Do Not cut open an alkaline battery, the chemicals inside are too corrosive.
Build a battery
You can build your own battery as well.
Salt water battery
A recently discovered battery uses aluminum foil.
Aluminum Air battery.
Measure your batteries.
Once you build these batteries you can measure them.
The two important things to measure are the open circuit voltage,
and the short circuit current.