In music theory, the octave is an interval that has twelve half steps
The octave requires that:
- The interval must be an octave interval (8 note names between the first and the last).
- The interval must have 12 half steps.
What does a perfect octave look like?
Here is an example of a melodic perfect octave (two music notes in a melody) and a harmonic perfect octave (in a chord):

Octave quality
Octave can only be perfect, it cannot be major, minor, diminished, augmented, (and so on...)
Inversion of the perfect octave
The inversion of the perfect octave is the perfect unison.
Here is an example of a perfect unison:

Musical examples of perfect octave
Beethoven, Symphony n°5
The first movement Allegro con brio from Beethoven's Symphony n°5 start with harmonic octaves:

This sheet music sample is an arrangement for piano by the famous pianist and composer Franz Liszt. Royalty free sound sample recorded in 1949 by the orchestra of the Paris Conservatory conducted by Carl Schuricht.(source)
You will notice the half-cadence
Interval identification game
You will find this interval in my Intervals identification game: