Octaves are notes that you hear as being “the same” in spite of their being higher or lower in actual pitch. (Technically, notes separated by an octave are in the same pitch class.) Play middle C on the piano. Then go up the C major scale (the white keys) and the eighth note you play will be another C an octave higher. The “oct” part of the word refers to this eight step distance up the scale.
From a science perspective, octaves are pitch intervals related by factors of two. When a tuning fork plays standard concert A, it vibrates at 440 Hz. The A an octave higher is 880 Hz, and the A an octave lower is 220 Hz. Any note with the frequency 2^n * 440 will be an A. It’s a central mystery of human cognition why we hear pitches related by powers of two as being “the same” note. The ability to detect octave equivalency is probably built in to our brains, and it isn’t limited to humans. Rhesus monkeys have been shown to be able to detect octaves too, as have some other mammals.