Getting at the electricity within a potato requires a copper and zinc electrode and alligator clips, although seeing the electricity also requires an electronic device like a digital multimeter. Push the electrodes into the surface of the potato about an inch apart. Attach one end of an alligator clip to one electrode the other end to the multimeter. Attach a different clip to the other electrode and to the multimeter. This will cause the multimeter to show around 1 to 1.2 volts, depending on the size and type of the potato.
Batteries give off electricity by taking advantage of the chemical reaction between two electrodes that are suspended in an electrolyte solution. For a commercially produced battery, this electrolyte solution might be sulfuric acid, potassium hydroxide, or a different compound, depending on the specific battery type. Negatively charged ions migrate towards one electrode and positively charged ions migrate to the other. The electrolyte solution connects the two gathering points in such a way that electrons flow between them from positive to negative (this is called a redox reaction). Ions are naturally formed, and potatoes contain enough juices and chemicals to drive the reaction.
Potatoes are not the only plant that can produce energy. Other foods that can produce electricity and which can be easily found at most grocery stores are: lemons, limes, oranges, grapefruit, pineapples, apples, bananas, carrots, tomatoes, zucchini, and cucumber. Similarly, zinc and copper are not the only possible electrodes to use, although alternate metals are harder to come by. Iron, steel, and nickel can work as electrodes and can be found at a hardware store. A gold earring stud can suffice as well, in a pinch.
Unfortunately, an individual potato battery does not produce enough electricity to even dimly illuminate a light bulb. Without modifications, there are no real applications. Nicely enough, it is simple to connect potato batteries in series and in parallel in order to increase the voltage and current so that they can power actual electronics. It only takes two potatoes connected in series to run a simple clock. In order to attach the batteries in series, connect the zinc electrode in one potato to the copper electrode in another. To attach them in parallel, connect the zinc electrode in one potato to the zinc electrode in another, and connect the copper electrode in the first potato to the copper electrode in the second. Given enough potatoes, one could run nearly any electronic system, although this is more expensive than using manufactured batteries.