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Polarity on a capacitor is because they're engineered to be higher capacitance in one direction by the way the dielectric and plates are configured inside the capacitor. This is typically to get a higher capacitance in a smaller package as the capacitance in one direction is sacrificed for capacitance in the other. These are used primarily on DC power supplies or in AC/DC circuits where the current in that particular part of the circuit is not going to change directions. Capacitors will still work if you hook them up backwards but the capacity will be lower and the voltage rateing is derated significantly. When I was a kid a friend of mine would hook up 16 volt electrolytics backwards to a 24 volt supply and we'd spend a couple hours blowing them up.
Above explanation is an invention. Capacitors which have + and - mark are polarized capacitors. Mostly they are electrolytics like aluminium or tantalum. Dielectric in these capacitors is thin oxide layer on electrodes and it has insulation properties only at right voltage polarity. If those capacitors are reversed polarized this dielectric will be destroyed and capacitor may explode because it get short circuited. Very thin oxid dielectric layer enables producing large capacitances on small volume because the capacitance of capacitor is inversely proportional to dielectric thickness.
Borber is correct. In addition, some thru-hole mounting type film capacitors have a band at one end of the capacitor. This band indicates the "outside foil", i.e., the terminal that connects to the foil layer that is exposed to the outside world. This is useful in some applications, where one end of the capacitor is grounded. The outside foil can act as a shield.
I would like to point out the bellow linked study I googled in a few seconds about reverse bias conditions on tantalum capacitors (please note I said at significantly derated values in my post) and stand by every word I said as being factual rather than an invention. I at no point recommended the use of polarized capacitors in a reverse biased state, merely that it was possible to use one in a reverse biased state under significant voltage derating.
Of particular interest to note is the conclusions section, save's a lot of reading. Points 1 and 2 say it all. Though I do admit I thought capacitance was affected as well which it is not.
Your point is well taken with respect to solid tantalum capacitors. I would also add the restriction that current limiting be employed. An interesting phenomenon occurs when solid tantalum capacitors are employed "back-to-back" to form a non-polarized capacitor. When the applied voltage is less than a critical value, which varies from process to process, the result is a capacitor with capacitance = C/2, as would be expected with other types. When the voltage is increased beyond the critical value where the "diode" effect (mentioned in the cited paper) occurs, the reversed capacitor acts as more-or less a short circuit, and the capacitance becomes C. No harm is done because the forward biased capacitor limits the current to a safe level.
Neither wet-slug capacitors nor aluminum electrolytics can be operated in the reverse-bias condition, even at low voltages, without severe and rapid degradation.