Role of wind shear in thunderstorm development
Wind shear plays a fundamental role in determining the internal dynamics of a thunderstorm, along with the organisation of a group of thunderstorms. Both speed and directional wind shear are important when considering thunderstorms. 
 
Role in discrete thunderstorms - Wind shear contributes to thunderstorm development by highly modifying the structure of the simple single cell conceptual model. This is that if a thunderstorm forms in an unstable environemnt with no wind shear, then a vertically-oriented updraft and downdraft will form. The updraft forms from unstable air close to the surface and the downdraft forms from the entrainment of cold, dry air and falling precipitation. As the thunderstorm matures, the downdraft becomes more dominant, choking the updraft and the supply of warm, moist air from the surface that is essential to sustain the thunderstorm. As a result, these thunderstorms usually last 30 - 45 minutes. They are most common in the tropics and are known as single cell thunderstorms, popcorn thunderstorms or airmass thunderstorms, because they typically form within a single airmass.
 
In order for a storm to last longer than this, the updraft and downdraft must be separated from each other: in other words, the storm must be tilted with height. This is where wind shear comes into play, as both speed and directional shear are able to separate the updraft and downdraft from each other, meaning that they don't interact with each other. Wind shear can generate a whole host of different storm types, such as multicell thunderstorms, which move as a cluster and can last several hours and the biggest and baddest of them all: supercell thunderstorms. These are characterised by deep rotating updrafts (mesocyclones) and produce the most severe weather, including large hail, very strong winds and tornadoes.
 
Speed shear is not only able to separate the updraft from the downdraft, but significant speed shear close to the surface can induce vorticity (a local turning motion) in the atmosphere, that generates rotating updrafts and ultimately the formation of tornadoes. Speed shear can tilt a thunderstorm downstream in the direction of the wind flow, meaning that the downdraft becomes tilted further downstream than the updraft, as illustrated in the image below. Directional shear can push the downdraft and updraft in different directions, also separating them. In general, speed shear throughout the depth of the troposphere promotes the development of multicell thunderstorms, whereas directional shear promotes supercells, since this encourages rotation. 
Effect of speed shear on a pyrocumulus cloud (from a forest fire) - this is analogous to a thunderstorm, however. The downdraft, where the precipitation would be, is clearly separated from the updraft, where the cloud has formed. This prevents the 'self destruct' mechanism that occurs when the downdraft and updraft are allowed to interact.
From https://www.reddit.com/r/askscience/comments/39u9mh/is_there_a_theoretical_limit_on_how_large_a/ â€‹
Speed shear close to the surface is what really enhances supercell thunderstorms however, because the shear produces vorticity that is ingested into the storm's updraft, enhancing the spin and rotation rate of the storm. From this low-level vorticity, a strong low-level mesocyclone is more likely to develop, which can produce tornadoes. A good way to visualise how shear produces this spin is to imagine a paddle wheel, with only a part of it exposed to a flow of air or water - the wheel will rotate, in the same way a mill wheel rotates.
A diagram illustrating how speed shear can generate a turning motion in the atmosphere. This motion is called vorticity. From http://www.srh.noaa.gov/jetstream/tstorms/windshear.htm
When an updraft is placed over this low-level shear, the rising motion causes the vorticity to tilt and the air to start rotating in the horizontal plane. This vertical vorticity helps contribute to (and increase) the overall rotation of the storm. Essentially, air that was initially rotating in a horizontal column, as show, above, is tilted by an overhead updraft, therefore becoming air that rotates in a vertical column. 
A lovely photo of a thunderstorm showing how a horizontally rotating column of air can be tilted into a vertically rotating column of air by an updraft. 
From http://www.wunderground.com/blog/24hourprof/archive.html?year=2013&month=05
Helicity - in simple terms, helicity is the extent to which the air exhibits corkscrew-like motion. Storm-Relative Helicity (SRH) has more direct links to thunderstorms, and the forecasting of SRH is an essential method by which forecasters can predict the liklihood of supercells and tornadoes. Going back to the picture above, the orange line is a streamline of the storm-relative wind. This is the component of wind that is moving directly into the storm and is calculated by subtracting the storm motion from the overall wind motion, and can be thought of as the movement of air relative to a stationary storm (even though storms are rarely stationary). In the picture, the air is rotating in a plane that is perpendicular to the storm-relative wind; this is known as streamwise vorticiy. Crosswise vorticity is the opposite of this, where the air is rotating in a plane that is parallel to the storm-relative wind. 
 
What is most significant for a supercell thunderstorm is the rate at which streamwise vorticity is ingested into the updraft, which contributes to the strength of the mesocyclone. This can be measured by the storm-relative helicity, and is computed by multiplying the storm-relative inflow by the magnitude of the streamwise vorticity. SRH is calculated by adding up all contributions from the ground to a pre-defined height - usually 1 km, or 3 km and is measured in m2/s2. Generally, any value over 100 m2/s2 for 0-1km, or any value over 250 m2/s2 for 0-3 km is deemed to be able to produce tornadoes.