A dutch roll is a yawing motion produced by an aircraft with greater lateral stability relative to directional stability.
If an aircraft makes an uncommanded bank in flight (turbulence), the aircraft will enter a small sideslip in the direction of the lower wing. If factors influencing lateral stability (dihedral) correct the bank before the factors resulting in directional stability (vertical fin size) correct the yaw, the aircraft will enter a sideslip in the opposite direction because the aircraft has yawed away from the direction of bank. The strength of the oscillation will usually decrease over time, but since it may occur over a long time span, it becomes very uncomfortable (it wouldn't have such an opportunity to stabilize in continuing turbulent conditions). The oscillations usually occur faster than the pilots reaction time, so pilot inputs will most often aggravate the motions causing "pilot induced oscillations." For these reasons, aircraft displaying properties of dutch roll must be fitted with yaw dampers (see below). Without yaw dampers, the quickest way for a pilot to stop the oscillations is to use full opposite control deflections, just long enough to stop the motions. [V-tail Bonanza owners make yearly sacrifices to their add-on yaw dampers because it improves handling and comfort 100%]
14 CFR 25 Airworthiness Standards: Transport Category Airplanes
Sec. 25.181 Dynamic stability.
(a) Any short period oscillation, not including combined lateral-directional oscillations, occurring between 1.2 VS and maximum allowable speed appropriate to the configuration of the airplane must be heavily damped with the primary controls--
(1) Free; and
(2) In a fixed position.
(b) Any combined lateral-directional oscillations ("Dutch roll") occurring between 1.2 VS and maximum allowable speed appropriate to the configuration of the airplane must be positively damped with controls free, and must be controllable with normal use of the primary controls without requiring exceptional pilot skill.
The yaw damper for older aircraft is a stand-alone unit with an accelerometer and servos, while newer a/c incorporate it into the basic control system.
Spiral Instability is the opposite of dutch roll; when directional stability is greater than lateral stability. After the wing drops, the aircraft yaws into the new relative wind before the wings level out. The accelerated movement of the higher wing produces more lift, increasing the angle of the bank and tightening the spiral. Since the spiral is easy to correct with intuitive control inputs, a/c are not required to have preventative measures installed (as is the case of the yaw damper for dutch rolls).