Polarized capacitors are only rated for voltage potentials in one direction. They like to collect charge in one polarity on their plates. A non-polarized capacitor such as generic ceramic types are capable of collecting charge in both positive and negative polarity (you can use them in circuits that have voltages that swing both above and below your zero/GND reference).
In this circuit you show, it is likely you are passing an A/C audio signal into the input. In this case, you are going to be feeding in a signal that swings to both positive and negative with respect to your GND points. You need a non-polar capacitor for this.
You'll damage a polar capacitor if you reverse the charge on it's plates. Fun fact! This is how capacitors explode, especially when used in circuits capable of sourcing high current.
Now you may ask "If I am passing in an AC audio signal, how come I can use a polarized capacitor on the output (the 250uF one)?". This is because the output of the LM386 can only swing between the rails of the power supply you provide it, in this case 0 volts to Vs. If you power this circuit with a 9v battery, or another DC source, you will not get a negative voltage on the output, allowing your high pass filter capacitor (250uf)to be polarized.
Fun fact: they list a 250uF polar cap symbol in the output because high value electrolytic capacitors are cheap and mass produced at such high capacitance values, whereas a 250uF non-polar ceramic or film capacitor is not something you can find very easily, if at all. Certainly not at a reasonable price.
Hope that helps!