What is an octave?

The word "octave" is related to "octopus", "octagon", etc -- in other words, eight. In music, an octave is 8 diatonic scale notes  higher or lower than  the note of the same name.

For example, the "A" note is always 8 notes higher or lower than the previous "A". The "A" above middle C vibrates 440 times per second, so the "A" an octave above it would vibrate 880 times per second, while the "A" below middle C would vibrate 220 times per second, and so on.

The human ear identifies these octave notes as being "the same" -- only higher or lower, so if a soprano sang A440 and a bass sang A110, the human ear would hear it as the same note -- just separated by pitch. That's why there are only 7 distinct diatonic pitches and only 12 distinct chromatic pitches, despite the fact that the piano keyboard has 88 keys. Each note is repeated over and over again, but at a higher or lower octave.