Double the frequency, same note
Go up an octave and the frequency doubles. A3 is 220 Hz, A4 is 440 Hz and A5 is 880 Hz. Notes an octave apart share a letter name and sound like the same note at a different height.
The word comes from the Latin for eighth. A major scale like C D E F G A B C has eight notes, with the first and the last an octave apart.
Equal steps in pitch aren't equal steps in Hz
Pitch perception is roughly logarithmic, so an octave is a ratio, not a distance. Going from 220 Hz to 440 Hz adds 220 Hz. Going from 440 Hz to 880 Hz adds 440 Hz. Both are exactly one octave. Human hearing shows the same thing. Its lowest octave, 20 to 40 Hz, spans just 20 Hz, and its highest, 10 to 20 kHz, spans 10,000 Hz.
That's why frequency tools use a log scale. The Frequency Generator's slider has its middle at 632 Hz, the halfway point between 20 Hz and 20 kHz in octaves, not at 10 kHz.
Twelve semitones, 1200 cents
Most keyboards use equal temperament, which splits the octave into 12 equal steps called semitones. Each semitone is split into 100 cents. A4 to A5 is 12 semitones, or 1200 cents.
Every semitone multiplies the frequency by the same amount, so one step up from A4 lands on 466.16 Hz and one step down on 415.3 Hz. The gaps in Hz differ, but the ratios match.
Octaves in bass, hum and harmonics
The 2nd harmonic of any note is an octave above it, so distortion and hum often show up an octave above the tone that causes them. Mains hum is the clearest case, since its fundamental is twice the power line's frequency.
Octaves also change how long a wave is. Each octave down doubles the wavelength in air: 14 feet (4.3 meters) at 80 Hz and 28 feet (8.6 meters) at 40 Hz. Long waves like these interact with the walls of a room, which is why bass can change so much from one seat to the next.