There are three ingredients that must be present for a thunderstorm to occur. They are moisture, instability and lift.
As a general rule, the surface dewpoint needs to be 55 degrees Fahrenheit of greater for a surface based thunderstorm to occur. A dewpoint of less than this is unfavorable for thunderstorms because the moist adiabatic lapse rate has a more stable parcel lapse rate at colder dewpoints. Dewpoints at the surface can be less than 55 degrees Fahrenheit in the case of "elevated thunderstorms".
Instability also decreases as low-level moisture decreases. Instability occurs when a parcel of air is warmer than the environmental air and rises on its own due to positive buoyancy. Instability is often expressed using positive CAPE or negative LI values. Instability is what allows air in the low levels of the atmosphere to rise into the upper levels of the atmosphere. Without instability, the atmosphere will not support deep convection and thunderstorms. Instability can be increased through daytime heating. Lift is what gives a parcel of air the impetus to rise from the low levels of the atmosphere to the elevation where positive buoyancy is realized. Very often, instability will exist in the middle and upper levels of the troposphere but not in the lower troposphere. Low level stability is often referred to as negative CAPE, convective inhibition, or the cap.
It is lift that allows air in the low levels of the troposphere to overcome low level convective inhibition. Lift is often referred to as a "trigger mechanism". There are many lift mechanisms. A list of many of them follows: fronts, low level convergence, low level WAA, low level moisture advection, mesoscale convergence boundaries such as outflow and sea breeze boundaries, orographic upslope, frictional convergence, vorticity, and the jet streak. All these processes force the air to rise. The region that has the greatest combination of these lift mechanisms is often the location that storms first develop. Moisture and instability must also be considered. A thunderstorm will form first and develop toward the region that has the best combination of: high PBL moisture, low convective inhibition, CAPE and lifting mechanisms.
The difference between a thunderstorm and a severe thunderstorm is the wind field. For a severe thunderstorm, the ingredients that must be present are moisture, instability, lift and strong speed and directional storm relative wind shear. Ideally, wind will have a veering directional change of 60 degrees or more from the surface to 700 millibars, upper level winds will be greater than 70 knots, and the 850 to 700 mb winds (low level jet) will be 25 knots or greater. Wind shear aids in the following: Tilting a storm (displacing updraft from downdraft), allows the updraft to sustain itself for a longer period of time, allows the development of a mesocyclone, and allows rotating air to be ingested into the updraft (tornadogenesis).
Severe storms also tend to have these characteristics over ordinary thunderstorms: higher CAPE, drier air in the middle levels of the atmosphere (convective instability), better moisture convergence, baroclinic atmosphere, and more powerful lift.