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Well, sometimes, when reading a schematic/diagram, it'll say what type of cap is most preferable and useful in that specific area. Sometimes you can tell what type of cap to use by the capacitance and voltage, ie; a 1200uf 80v cap, would quite obviously not be a film or ceramic type but an electrolytic. Electrolytics are pretty much always the ones with the higher capacitance, around 10uf and up in most cases. Film and ceramic are used usually for 10uf and under, these types can quite often be interchanged with no damage done. Of course, there are electrolytic caps lower than 10uf and film types over 10uf, but there is a limit (if size is an issue). For example, when working with ps's, usually the caps will be electrolytic or ceramic, audio projects will usually use film types(when asking for lower capacitances) but in this situation is not recommended to exchange film with ceramic. I really havent read up on what cap types are used for what but am "writing" from experiance.
Well as far as I can remember, there is no difference between A and B, but as for C, this kind is almost always an electrolytic, because these types have + and - cathodes (- is usually the one marked guess it's just to waste less ink of the shrink plastic) it may be another kind, there are one or two other types of capacitors that have specifics like this, but if you see this on a schematic it's safe to say it's electrolytic.
Many projects and applications are somewhat insensitive to the differences between capacitors so what's been said already is usually sufficient. In some applications the differences are important.
Some characteristics of capacitors that might matter:
Intermal resistance (ESR, among other things) - the perfect capacitor has none and the different types have greater or lesser amounts. I think electrolytics have fair amounts - silver micas have little.
Temperature can profoundly affect the capacitance - ceramics are characterized by this so that designers can predict what will happen - often one type is used to offset the temperature affects of something else in the circuit.
Voltage can affect the capacitance - again, different capacitors show different amounts of this. I know data is published for ceramics.
Heating as a result of AC current flow has an effect on capacitors. If significant amounts of RF current are expected to flow then heating of even a small capacitor can occur - even if the working voltage is not exceeded (frequency dependent).
Frequency of operation does impact capacitance (yes, reactance too).
Inductance is there in small amounts though in the perfect capacitor there would be none.
As a rule, experience or recommendations based on application usually is a good enough guide to which type to use. It's usually not necessary to agonize too much though some attention is required. I know in a radio receiver you might carefully choose the type in the frequency dependent circuits to get the least drift.