Stable and unstable airmasses - what does all this mean?
First, to visualise what 'stable' and 'unstable' states mean in a physical sense, stand a round pencil on end on a level surface. From Newton's First Law of Motion, it will remain upright until a force is applied. Once displaced, the pencil falls over, failing to pass through its original (upright) position. This is the UNSTABLE state. Now lay the pencil on its side, at the bottom of an incline. Displace the pencil slightly up the incline, then remove the force of displacement. The pencil will return to its original position. This is the STABLE state.
In the atmosphere, whether air that is displaced does so in an unstable or stable environment depends upon the vertical temperature profile of the air -- its lapse rate -- and upon the moisture content of the parcel. These differences are fundamental to understanding why clouds take up the form they do.
In the atmosphere, when a 'parcel' of air moves vertically upwards (or downwards), it cools (upward motion), or warms (downward motion), in accordance with thermodynamic rules ... if the air is unsaturated (air temperature > dew point temperature), the cooling/warming will be at a rate of 3 degC per 1000 ft (or 10 degC per 1 km): This is known as the Dry Adiabatic Lapse Rate/DALR; If the air is/becomes saturated (air temperature=dew point temperature), this rate is roughly halved in the lower troposphere, due to the release of latent heat upon condensation. This rate is known as the Saturated Adiabatic Lapse Rate/SALR.
Such ascent/descent is said to be adiabatic, which means that the energy/heat changes are confined to that particular parcel.Provided the parcel is warmer (less dense) than the environmental air through which it is passing, it is buoyant, and rises. If the parcel is colder (denser) than ambient air, then it will descend, or try to descend. Because the rates of cooling (ascent), and warming (descent) of individual parcels are fixed, the important variable is the overall lapse rate (i.e. the rate of change of temperature with height) of the atmosphere. On average, this is 1.98 (call it 2 degC) per 1000 ft, or 6.5 degC per 1 km in the troposphere, but this average conceals a wide variety of cases which are important in meteorology.
Where the temperature falls off slowly with height, or indeed rises, e.g. in a slow moving anticyclone, or a tropical maritime airmass, then an air parcel subject to lifting/adiabatic cooling will readily find itself colder than its surroundings ... denser ... and try to return to its original position: The air is ABSOLUTELY STABLE. Where the temperature falls off quickly with height, e.g. in a cold/polar air mass over NW Europe in late winter/spring, then an air parcel subject to ascent, although cooling, may still find itself warmer/less dense than its surrounding air ... it will be buoyant, and tend to rise further: the air is ABSOLUTELY UNSTABLE.
Problems arise when, on ascent, the dew point of the air is reached, and the rate of cooling is therefore less - it follows the SALR figure. If, however, the parcel is still warmer/less dense, then it will continue to rise, and the condition of the air is said to be CONDITIONALLY UNSTABLE .. i.e. conditional upon whether the parcel is saturated or not. This is by far the most common situation in the 'real' atmosphere, accounting for some 65-70% of situations taking the troposphere as a whole.
Stable airmasses generally imply the absence of 'free' vertical motion, and any ascent that does occur must be forced, i.e. frontal (dynamic or mass) or orographic (mechanical) ascent, and the cloud structure is essentially layered. (NB: Forced ascent comes about in several ways: frontal ascent due to large-scale air motion within frontal systems, with of course adjacent descent; convergence into an area of low pressure - the converging air can't go down near the surface - it has to go up; and topographical forcing, that is, air is forced to rise over major upland ranges. )
Unstable airmasses imply free vertical motion (given an initial trigger action), and the cloud structure is 'heaped' or cumuliform. If the vertical motionis vigorous and deep enough, and there is sufficient moisture, then heavy showers/thunderstorms are likely. (NB: Trigger action: method of causing air to rise initially, which in the lower troposphere include not only the 'wide-area' triggers noted above under stable conditions, but also smaller/mesoscale mechanisms such as differential heat response between land and sea, coastal convergence, etc.)
For more information on these subjects, see a good textbook on meteorology, for example, Essentials of Meteorology:(Taylor and Francis/D.H.McIntosh and A.S.Thom).