Stability and instability
When it comes to forecasting thunderstorms, atmosperic stability and instability is key. The atmosphere is a fluid and as a result, it is influenced by buoyancy. Less dense air is more buoyant and so will rise through the overlying layers of the atmosphere in order for the atmosphere to reach thermodynamic equilibrium. When air is heated, it expands, therefore becoming less dense and more buoyant. Heating of the air usually occurs at the surface, which results in thermals and cumulus clouds developing in the summer. There are several methods to measure the amount of energy available to a parcel when it rises through the atmosphere. 
Lifted index - this is calculated by taking the difference in temperature between a parcel of air that has risen adiabatically from the surface (that is to say, initially along the dry adiabat below the cloud base and then along the saturated adiabat), and the environmental temperature, at 500 hPa. 
Lifted Index
         Positive number                   Increasing stability
         0 to -4                                     Slight instability
        -4 to -8                                     Large instability 
        -8 and below                          Exceptional instability
CAPE - convective available potential energy. This is a measure of the energy a surface parcel of air would have if it was raised, allowed to condense and continue rising to the maximum parcel level. It is represented as a potential energy - that is to say that it has not yet been used by the parcel. Only after convective clouds or thunderstorms form is this energy "used up" - in fact, it is converted from potential to kinetic energy. It is calculated by finding the positive area on an atmospheric sounding diagram between a parcel of air that has risen adiabatically from the surface, and the environment, i.e. the area where the parcel temperature is warmer than the environment through the vertical slice of atmosphere. Because it is a measure of energy, CAPE is given the units of joules per kilogram (of air). This is the amount of energy that can be expended by 1kg of air at the surface if it is allowed to rise vertically, forming an updraft.
         1 - 500                                     Positive 
         500 - 1500                              Slight
        1500 - 2500                             Large CAPE 
        2500 +                                      Exceptional CAPE 
In general, higher values of CAPE lead to faster updraft speeds, which result in faster storm development and the potential for more significant severe weather, such as large hail, lightning, rainfall and strong wind gusts. Of course, these tables should be taken with a pinch of salt - how unstable the atmosphere is when the threshold values (in the tables) of CAPE and lifted index are reached is subjective, and depends on where you live and what time of year it is. For example, 1000 J/kg of CAPE is very large for the UK at any time of the year, although it is fairly modest for Oklahoma in the spring and summer.
CIN - convective inhibition. This is essentially the opposite of CAPE, and is a measure of the amount of energy a parcel of surface-based air would need to rise through a region of stable air - also known as a region of warmer air, or a temperature inversion. As opposed to CAPE, it is calculated by finding the negative area on an atmospheric sounding diagram between a parcel of air that has risen adiabatically from the surface, and the environment. This is the area where the parcel temperature is cooler than the environment. Although CIN has the ability to prevent thunderstorm development, it can also increase the CAPE that is allowed to build during the day, essentially bottling up the surface air and only allowing it to be released explosively in the late afternoon when the surface temperature has risen enough to overcome this temperature inversion, allowing rapid thunderstorm formation. This "cap" as it is known by meteorologists, plays a fundamental role in severe thunderstorm development in Tornado Alley - and indeed anywhere in the world. Both stability and instability can be enhanced or reduced by large-scale synoptic processes, such as the passage of a front or a depression.
A stable layer of air often develops above the surface on cold, clear nights in the autumn and winter. As the surface radiatively cools at night, the first few metres of air above it cool through conduction, but the air above this is insulated from below, leading to a warm temperature inversion and the formation of fog which can become trapped between this layer and the surface as the air cools and reaches its dew point. 

This sounding from Dodge City, Kansas, from the 20th July 2015 displays both stability and instability. The black line represents the temperature of an air parcel rising from the surface. From the surface to 660 hPa, the parcel is in an area of CIN. From here to 170 hPa, the parcel is in a region of positive buoyancy; CAPE. Above this, the parcel is in a region of CIN once again - it has reached the stratosphere.