There's a few different types:
Analog octavers (Boss OC-2, EBS Octabass, MXR Bass Octave Deluxe, Aguilar Octamizer, Foxrox Octron etc) - they all run your bass's signal through some filters, into a divide by 2 circuit. The result is a square-ish waveform one octave down from your bass's fundamental frequency. This is then multiplied by your original bass frequency to give something which sounds natural but sounds an octave down. Extra filtering removes some of the effect of the squarewave. Some have an extra divide by 2 to give two octaves down. Some (Pearl OC7, Octron) have an octave up - this is a half-wave rectifier circuit that flips the negative parts of the bass waveform positive, faking an octave up.
Because of the divide by 2, these pedals need a strong fundamental frequency in your bass tone. When they don't get that, the tracking can suffer - notes warbling between two pitches, or the octave dropping out suddenly. You can't play chords or expect super low notes to be tracked. However they have a great synthy tone that people love.
Subharmonic generators (DOD Meatbox, Mantic Density Hulk, DBX 120, Ashdown amps) combine an octave down effect with eq to give a big low end boost. The octave down is similar to an analog octaver, but uses a different chip (now obselete, but Mantic must be getting them somewhere). The octave effect is not as separate as an analog octaver - it merges with your bass tone more. The Meatbox also has EQ knobs around 30 and 60 Hz. The circuit also distorts when the EQ knobs are turned high.
They still have the same limitations as an analog octaver though - no chords, notes too low etc. Obviously, you need suitable amplification to reproduce the low frequencies. Not all PA guys love the huge bottom end they produce as it can destroy their subs!
Digital octavers (EHX Pog, Hog, Digitech Whammy, Boss PS-3) these convert the audio into a digital signal, perform a pitch shifting algorithm on it, then convert it back to an analog signal. The pitch shift can be controlled more than analog octavers - usually any interval within a 2 octave range or so. With the Whammy you can use the pedal to slide pitches up and down.
They can usually track a wider range of pitches, chords etc. The tone is closer to your bass's original than an analog octaver, but there may be artefacts like warbling, a delay before the note etc. Some people love the sound of them, some don't.
Anything I've forgotten?