How it works
A standard chronograph requires three pusher actions to reset-and-restart: press to stop, press to reset, press to start. A flyback chronograph compresses these into one press: while the chronograph is running, pressing the reset pusher (typically at 4 o'clock) immediately stops the centre seconds hand, snaps it back to zero, and restarts it from zero in one continuous motion. The chronograph briefly disengages and re-engages the clutch in a single timed sequence.
Why pilots wanted it
Aviation chronographs in the 1930s-50s were used for in-flight timing: the pilot starts the chrono at a navigation waypoint, then resets-and-restarts at the next waypoint without the brief 'unmeasured' window that a stop-reset-start sequence would create. The flyback mechanism removes that window. Breguet Type 20 (French Air Force, 1954) and the Type XXI are the canonical flyback pilot chronographs.
Modern flyback chronographs
Most modern flyback chronographs are higher-tier in-house movements: Patek CH 29-535 PS, Audemars Piguet Cal. 4400, TAG Heuer Heuer 02, Zenith El Primero 405 (some references). The mechanism adds 5-10 components over a standard chronograph and requires more precise component fitting; flyback chronographs typically retail 20-40% above their non-flyback siblings.
When it actually matters
For real flight or motorsport timing, flyback is genuinely faster and more accurate. For everyday wear, you almost never need it; the chronograph is mostly used for casual elapsed-time measurement where the 3-press sequence isn't a problem. Flyback is now mostly a connoisseur-spec marker (the dial often labels it explicitly: 'Flyback' or 'Retour-en-vol'). See wiki: flyback chronograph.