Octave
An octave is the interval between two notes twelve semitones apart, such as one C and the next C up, where the higher note vibrates at exactly twice the frequency of the lower.
What does an octave look like on real notes?
On a piano, count twelve keys up from any C, black and white together, and you land on the next C. That is one octave.
The frequencies double at each step. The A that orchestras usually tune to, A4, is 440 Hz. The A an octave above, A5, is 880 Hz. The A an octave below, A3, is 220 Hz.
On a guitar, the 12th fret of any string plays the same note as the open string, one octave higher, because pressing there halves the length of the string that vibrates.
The numbers after note names count octaves. In the most common system, middle C is C4, and each new number starts at C: B3 is the note just below C4.
How does an octave sound?
Like the same note, higher or lower. The two notes blend so completely that they are easy to mistake for one, which is why music gives them the same letter name. When men and women sing a tune “together”, they are often an octave apart without noticing.
This is also why the twelve note names are enough: past B the names start again at C, one octave up.
Why is it called an octave if it has twelve notes?
Because the name counts letter names, not semitones. Walk up the C major scale, C–D–E–F–G–A–B–C, and the second C is the eighth note, counting both ends. Octave comes from the Latin for eighth.
In between there are twelve semitones. Both counts are correct; they measure different things. Mixing them up is the most common slip in interval arithmetic, and it is why a ninth is a second plus an octave, not a third.
Try it: Notes and Semitones has you count semitones on the keyboard, octave included. To find every octave of a note on the guitar, see How to memorize the fretboard.