CIFK stands for Convective Instability of the First Kind. Convective instability is a situation in which thunderstorms will occur when air rises through unstable air. One factor that promotes instability is warming air near the surface. Another factor is cooling air aloft in the middle troposphere. Very warm air under cold air is an unstable situation. Think of a pot of boiling water. The air rises from the stove toward the ceiling since warm air rises through the surrounding cooler air since the warmer air is less dense. CIFK only deals with the temperature profile. CIFK becomes more significant as the air near the surface warms (i.e. daytime heating, warm air advection) and the air aloft cools (i.e. evaporation, cold air advection, adiabatic lifting).
CISK stands for Convective Instability of the Second Kind. Not only does a changing temperature profile lead to more or less instability. Instability is also influenced by the moisture profile. Instability will increase further when the dewpoint near the surface increases. The addition of moisture makes the air less dense near the surface but more importantly releases more latent heat when air rises and condenses into precipitation. Air that is less dense and is warmer due to latent heat release will produce a more significant convection. Severe thunderstorms often have an environment with warm and humid air near the surface with cool air aloft.
In conclusion, instability is influenced by the temperature and moisture profile. The three factor that will increase instability are warming air near the surface, cooling air aloft and increasing dewpoints near the surface.