Stability in the atmosphere refers to a state of equilibrium reached in the atmosphere when a mass of dry air which is rising in form of a wind has a lapse rate that is greater than that of the surrounding air i.e the environmental lapse rate of an air mass is less than the dry adiabatic lapse rate.
The rising air mass loses its temperature ultimately becoming cooler and denser than the surrounding air mass. It then sinks back to the ground level unless when some external force is at work.
Instability in the other hand is a state of unstable equilibrium of the atmosphere where the environmental lapse rate of an air mass is greater than the dry adiabatic lapse rate. A surface pocket of unsaturated air when heated, will rise and cool at the dry adiabatic lapse rate and because it is warmer than the surrounding air mass, it will continue to rise.
Due to the fact that the surrounding air is denser than the rising air mass, it will force it up to greater heights where it cools from. High instability leads to the formation of cumulo-nimbus clouds, stratus, and cirrus clouds.
The strato-cumulus and cumulo-nimbus clouds are associated with intense rainfall and thunderstorms.
The cirrus and stratus clouds give rise to clear skies / sunny weather conditions
The alto-cumulus and alto-stratus clouds give rise to light drizzles and unstable windy conditions
High humidity is formed within the atmosphere