Polyphonic refers to a synthesizer that is able to play two or more sounds at the same time. A digital keyboard is polyphonic as it allows the user to play chords of simple or complex nature. The number of notes are sometimes described as voices (4-voice, 8-voice).
Polyphonic synthesizers allow users to quickly build up complex layers of sound very easily and quickly. The more voices you have available in your synth, the more possibilities there are for experimentation and creative expression.
Over the years, technology has been able to give synthesizers more and more polyphonic capabilities. Common models show anything from 4 to 128-note polyphony. 128-note means it would be capable of playing 128 sounds at any one time.
While you might think that every note played is a single sound, this is not always the case, as the individual sound might be constructed of several other sounds. In this case, the number of polyphony notes being used in one note would be greater than one voice.
You might also think that humans are only capable of playing a certain number of notes with two hands, but synthesizers allow notes to keep on playing after the key has been released. For instance, the synth can still be processing the sound using sustain effects and pedals.
Polyphonic synthesizers tend to reduce the quality of the overall sound, as opposed to using monophonic synthesizers. The processing power required and digital recreations of instruments will always reduce the natural characteristics of the sound.