Atmospheric stability tells you how likely it is air will rise and form clouds and precipitation. Unstable air is likely to rise and form tall clouds and often precipitation. Stable air means any air that rises up will sink back to where it came from.
Why do I care? Stability dictates where and if inclement weather will occur.
I should already be familiar with: What Drives Weather
Figure A. Diagram describing stable, unstable and neutral air. (Image from Columbia University)
Figure A shows images of air parcels. If the temperature of the air parcel is greater than the temperature of the surrounding air, the parcel will rise and instability will occur. If the temperature of the air parcel is less than the temperature of the surrounding air, the parcel will sink, because it is heavier than the surrounding air, and stable conditions will persist. If the temperature of the parcel and the temperature of the surrounding air are equivalent, the parcel will remain where it is located in the atmosphere until conditions vary.
Stability is a measure of the likelihood that air will rise and form clouds. It hinges on the basic premise that hot air rises and cool air sinks. When conditions are stable, if air from the ground is blown upward by an updraft or lifted over a mountain, the lifted air will be cooler than the air around it and will sink back down to the ground. When conditions are unstable, if air from the ground is pulled up by an updraft, the lifted air will be warmer than the air around it and will continue to rise. As the air continues to rise it will eventually reach a level where the moisture will condense and form clouds and sometimes precipitation. Weak instability generally means any clouds that form will be fair weather clouds and produce no precipitation. These clouds are relatively low in height. Highly unstable conditions mean the clouds that form can become strong thunderstorms with heights up to 10 miles deep.