The octave is an interval (distance between two notes) embracing eight notes (for example, A-B-C-D-E-F-G-A).
An octave is also a note that has exactly the double number of vibrations of another note. Example: if one note has 100 vibrations per second, its octaves will have 200, 400, 800 etc. vibrations per second.
Playing "octaves" also means to play the two notes together.
Because of this ratio between the notes, it's very easy to spot when the two notes of an octave are not perfectly in tune.