If you play a real piano, each note would be one voice. Play a triad and you're using 3 voices. Real pianos have 88-voice polyphony, meaning that you can play 88 unique notes at the same time. Polyphony = number of playable voices at once.
On a keyboard, you have electronics creating the notes, not physical strings, but the idea is similar. If you're in a program mode (not some other mode where you're layering programs together), each key you press will take up at least 1 voice. They can use more than one if the sound you're using has multiple parts to it - using more than one oscillator, or using overlapping velocity layers (i.e if you press the key really lightly, trumpet sample A will play; if you press it really hard, B will play; if you press in the middle, both A and B will be mixed). In general, most keyboards would use multiple voices for this, though some keyboards might use only one per program per key, no matter how complex the program is.
Manufacturers put a limit on the polyphony to keep costs down. Some synths let you layer up to 16 sounds at once, and each sound can take up more than one voice, plus you can play a large number of keys by using a sustain pedal. Say you press 10 keys at once and you've got a layer of 16 things, each using two voices: you'd be using 10 * 2 * 16 = 320 voices, and with sustain it can go up from there, so manufacturers obviously need to limit things.