Stability by design
Transport aircraft are inherently stable. They are designed with a positive static stability, meaning they tend to automatically correct for variations in their attitude and return to neutral. E.g., the wings are not built horizontal; they make an angle (dihedral or anhedral) to return to zero bank after a wind gust. Without this roll stability, the aircraft could continue to roll after this motion has started.
However, this stability mustn't make piloting difficult, e.g., too much roll stability would require excessively large efforts on control surfaces. The compromise is a design choice, e.g., fighter aircraft have less stability in order to facilitate fast maneuvers, while transport aircraft are less agile, but more stable.
With less stability, safe piloting may require computers (e.g. to move canards). The same principle of aided flight is also used on more stable aircraft, to relieve the pilot from constantly adjusting attitude, or to react faster than the pilot, e.g. for gust compensation.
When you feel instantaneous reaction, it's likely an action of these computers.
Flight computers for additional stability and protection
Additional stability isn't natural, but managed by computers distinct from the autopilot maintaining programmed route and altitude, dedicated to the actuation of control surfaces. This system is what is generally called fly by wire.
Computers receive pilot orders, calculate a combination of surface movements, within predefined safe limits, send electrical signals to actuators, ensure the calculated changes are carried out, then maintain the commanded attitude.
For attitude keeping, the principle is to detect accelerations and orientation changes on each of the three axes. This is usually done by the inertial reference system which also computes the aircraft position. These inertial data are used by the flight computers.
One example of such computer is the yaw damper, which counteracts both a yaw and roll oscillation in order to prevent a flight dynamic mode known as Dutch roll which can be catastrophic if left uncontrolled, and has been the cause of accidents.
On Airbus aircraft, attitude correction is controlled by seven computers (ELAC, SEC, and FAC, see details in Airbus control architecture - where does the actual autopilot live?) which can reconfigure themselves if one or more fail, to continue providing protection and stability as much as possible. They work regardless of who pilots, human or autopilot.
These computers set the attitude commanded by the pilot, and maintain it when input has ceased. They also ensure the commanded attitude and the commanded rate of change are within the possibilities of the aircraft (flight envelope protection). On the schematics above there are two mechanical links which are last resort manual controls to land the aircraft in case the fly by wire system has failed to the point orders cannot be transmitted.
For bank stability, this is the way Airbus A320 family works. From manufacturer FCOM: