Mesocyclones first appear in the midlevels (say, around 10000 feet to 18000 feet) of thunderstorms growing in an environment of sufficient vertical shear (i.e., in the presence of favorable jet stream winds). Meteorologists estimate this favorable vertical shear by subtracting the surface wind direction and speed (combined, called the surface wind vector) from the wind vector at 6 km (~18000 feet, approximately the 500 mb level). This is sometimes called the "deep layer shear".
When the magnitude of this difference is around 35 knots or greater in an environment (and, usually, in the lower level winds are nearly at right angles to the winds at 18000 feet or so but this is not necesary), the shear is thought to be favorable for midlevel mesocyclone development. (Note: the 500 mb wind can be used as a rough estimate of the deep layer shear.) Such vertical shear also ensures that the storm will be long-lived enough to give the rotation time to develop because the stronger upper winds prevent precipitation from falling through and suppressing the updraft.
Since the jet stream is almost always present over the United States (except in mid summer, usually), it might be assumed that the deep layer shear is almost always favorable for storms to develop rotating updrafts. However, only in areas in which the surface layer wind is at a substantial angle relative to the 500 mb wind will the inflow air to developing thunderstorms possess substantial spin due the vertical shear. A measure of the degree to which the surface wind streams possess this spin is the STORM RELATIVE HELICITY (SRH or SREH). This is normally calculated for various layers from the ground to about 1 km or up to 3 km. Helicity (from the word helix) is a measure of how "helical" the updraft of a growing thunderstorm will be.
When the updraft develops rotation through a deep portion of the mid section of the storm, and if that rotation persists for periods on the order of 15 minutes or more, the storm is classified as a "supercell." Despite popular myth, the definition of the supercell contains nothing about the storm size or depth, or whether the storm produces a tornado. In fact, studies have shown that only 10% or so of supercells produce tornadoes. However, supercells account for a disproportionate fraction of the severe weather reports received by SPC (meaning, most thunderstorms are not supercells, yet most severe weather reports are associated with supercells).
All supercells that directly produce tornadoes have mesocyclones that appear to extend to the ground from the midlevels of the storm; this is due to the vertical shear in the lowest layers of the atmosphere.1
Studies have shown that supercells developing in environments in which 0-3 km SREH values of 150 or greater are observbed often associated with weak tornadoes (F0). Values of 250 or greater suggest to meteorologists that if tornadoes occur they are liable to be associated with moderate (F1) to considerable (F2) damage.
These are just general limits---some studies have shown that SRH values do not need to be this great if the buoyancy (as measured by CAPE) is great, while situations of relatively low buoyancy (low CAPE) can be mitigated by very strong vertical shear. In order for helicity to be important, however, there must be some buoyancy. So meteorologists often look at the combination of helicity/deep layer shear and CAPE in a given area. Keep in mind that there are other factors that relate to whether or not a supercell will produce a tornado in the so-called "supercell cascade." But these other factors do not become important unless the deep layer and low level shear develops in a proper manner, as outlined above.
1Recent research suggests that low level positive shear (0-1 km positive shear) may be a more direct measure of the potential of the low level rotation to develop.