Capacitors
The simplest conceptual model for a capacitor is called the "parallel plate capacitor." It's just what it sounds like: two flat pieces of metal that are lined up with their faces parallel. There is no electrical connection between the two plates, so when one plate is hooked to the positive terminal of a battery and the other is hooked to the negative, current does not flow from one plate to the other. But electrons are pushed to the surface of the plate connected to the negative terminal. In response, electrons in the opposite terminal are pushed away from the surface -- because like charges repel each other. So current flows until the capacitor can't take any more and the capacitor ends up storing electric charge.
The Capacity
The amount of charge a capacitor can store is its capacitance. The amount of charge a parallel plate capacitor can hold goes up as the area of the plates goes up and as the spacing of the plates goes down.
There's another way to increase the capacitance of the parallel plate capacitor: put a non-conductive dielectric material between the plates. The structure of the atoms within the dielectric material distorts when the capacitor is charged, which has the effect of reducing the voltage necessary to maintain a certain amount of charge. Put another way, a dielectric capacitor has a higher capacitance than the equivalent capacitor without a dielectric.
Geometry
Just about every electronic device has capacitors built in. It would be rather inconvenient if a circuit needed to be filled with metal disks, so the typical capacitor is constructed in more compact fashion. In principle, they're built more or less like this: a sheet of tin foil is laid down, then a thin sheet of dielectric plastic is laid down on top, then another sheet of tin foil. Then the whole thing is rolled up like a jelly roll.
Oil-filled Capacitors
Capacitors have a number of uses. One of them is for storing current to be released in a burst. For example, fluorescent lamps need a burst of current to start, and capacitors usually hold that current and release it. High-intensity discharge lamps are similar, but they need the current to be injected at a high voltage. When a high voltage is put across the metal foils that are rolled up next to each other in a capacitor, they can ionize the air between the plates, causing a spark to pass between the different sides of the capacitor. At best, this reduces the amount of charge in the capacitor, and at worst, it ruins the capacitor, sometimes with a spectacular explosion. That's why many high-voltage capacitors are filled with oil: to reduce the chances of arcing.
Dry Film Capacitors
The oils that fill high-voltage capacitors are a problem. In the old days, these capacitors were filled with oils containing PCBs, now known to be an environmental hazard. Even without the environmental hazard, oil-filled capacitors are heavy and pose a leakage hazard. That's where dry film capacitors come in. They have a special dielectric material between the rolled metal foils, a dielectric that resists high-voltage breakdown. Dry film capacitors are alternatives to oil-filled high voltage capacitors. Dry film capacitors are lighter, smaller and won't leak.