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Batteries store energy and so do capacitors. However, the principles are much different, and batteries can store much more energy.
Batteries actually store chemical energy. One can make a battery, for example, by placing a piece of zinc and a piece of carbon (such as the graphite in a lead pencil) into a lemon (which serves as an electrolyte). The battery will work until the zinc has been used up (providing the electrolyte has not dried out). However the battery is made, the total energy E that it can produce is the voltage V of the battery multiplied by the current I it produces multiplied by the time t. (E = VIt) The product It is the total amount of electric charge Q that moves through the wires when electric current flows, to the energy is E = VQ.
In a capacitor, the energy comes from the fact that the stored charge on one plate repels like charge on the same plate and attracts unlike charge on the other. When an external circuit is provided, the charge flows from one plate to the other, each tending to neutralize the other. If half the charge becomes neutralized, the voltage is half of its starting value. (In a battery, the voltage remains constant instead.) In terms of the charge Q stored on the capacitor and the starting voltage V, the energy E = (1/2)QV.
There are many ways to make capacitors, but the important considerations are the surface area of the plates and the distance between them. If you want a capacitor to be able to hold a large voltage V, the distance between the plates must be large; however, the amount of charge that can be stored at a given voltage is diminished. Overall, the energy that can be stored in a capacitor depends mainly upon the overall volume of the capacitor.