Here are the most common stability / instability terms you will run across and the definition.
Potential Instability-- Instability caused by mid-level dry air being advected over the top of PBL warm/moist air. If the troposphere is forced to lift (i.e. by a front, jet streak, vorticity, etc.) the warm/moist air will initially rise at the moist adiabatic lapse rate (avg. of 5 C/km) while the dry air will cool at the dry adiabatic lapse rate (9.8 C/km). Over time, the temperature lapse rate (rate of cooling with height) increases.
Convective Instability-- The same as potential instability
Absolute Stability-- Any layer in the atmosphere where the rate of actual temperature decrease (or increase) with height is less than the cooling rate of the moist adiabatic lapse rate. The most stable layers are inversions (a.k.a. cap) where temperature increases with height.
Absolute Instability-- Any layer in the atmosphere where the rate of actual temperature decrease with height is greater than 9.8 C/km (rapid cooling with height). On a sounding these are termed superadiabatic lapse rates. They most commonly occur at the surface during strong solar surface heating.
Conditional Instability- Any layer in the atmosphere where the rate of actual temperature decrease with height is between the moist and dry adiabatic lapse rate. The air is unstable if saturated but stable if unsaturated. Saturated air cools less with height due to latent heat release thus allowing the parcel to be warmer than the environment if lifting occurs in a conditionally unstable environment.
Latent Instability-- Instability caused by the release of latent heat. The more latent heat that is released, the more a parcel of air will warm. If the PBL is very moist and humid, the moist adiabatic lapse rate will cause cooling with height of a rising parcel of air to be small (perhaps only 4 C/km) in the low levels of the troposphere. A low pressure with an abundant amount of moisture to lift will have more latent instability than a low pressure that is surrounded by dry air. Often low pressures will intensify once they get to the east of the Rockies because more moisture becomes available to lift. A Nor-easter is a classic example of latent instability. Warm and moist air from the Gulf Stream or Gulf of Mexico increases latent instability.
Notice that all the stability / instability terms deal in some way with the release (or lack of release) of latent heat.