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With The piano - it will most likely always be in tune so thank god for that. Octaves are simply a tone that has the exact same Note. A Tone is a "Pitch" created by vibrations that beat at a certain rate - (beats per second) The "Tone A" vibrates at 440 Beats per second - So ANYTHING that vibrates at 440 beats per second will sound out the "Tone of A"
The OCTAVES of the "Tone of A" are the numbers that are multiples of (440)
An Octave Below 220 is very low - it would have to be loud enough to tear your ears up to hear its tone - 
220 Beats per second Octave below A
440 Beats per second Center A
880 Beats per second first Octave A
1,320 Beats per second second Octave A
1,760 Beats per second third Octave A 
2,200  Beats per second fourth Octave A 
2,640  Beats per second fifth Octave A
3,080  Beats per second sixth Octave A
3,520  Beats per second seventh Octave A
3,960  Beats per second eighth Octave A

If you are running a lawn Mower and the engine is running at 3,080 Revs Per Second - you will hear an "A Tone"
If you have a fat truck and its Idling at 880 Revs per second - You will hear a very Cool "tone of A"
It doesn't matter what the vibration is - its the rate at which it Vibrates that determines the pitch.      

  There are only twelve tones on the Piano - The rest are merely octaves. The piano's Octaves Jump out at you - Notice the "Pattern" of Black Keys and White Keys. 
The Two black keys and Three black keys  is the Pattern that sticks out the best - the first black key on the Two Black keys part of that pattern are ALL Octaves of each other, So ANY note will be the same in correlation to the pattern - look at the Diagram Below and practice the sounds of Octaves - by just playing the octaves up and down the Piano - Any note at all

You should be able to Play octaves with ONE hand (Thumb and Pinky) - this is a very typical Goal that all Piano players can play Octaves - its a good stretch if your not use to it or have little hands! Practice playing octaves with Both hands!