Multicell thunderstorm
The next up in the thunderstorm pecking order is the multicell thunderstorm, which is probably the most common form of convection in the midlatitudes. It is made up of many cells, where each individual cell goes through its own life cycle - a series of evolving cells with each one, in turn, becoming the dominant cell in the group, but the group as a whole persists.
 
These storms are likely to occur in an environment with moderate to large instability and low to moderate vertical wind shear, but importantly, with little turning - meaning that the shear needs to be mostly speed shear, rather than directional shear. As a result of the wind shear, the storm cells do not become vertically stacked and so last for longer and can produce more severe weather than single cells. The cold outflow from the evaporatively cooled downdrafts combines from each cell to form a much larger and stronger gust front at the surface. Rapid and forced ascent of warm air at the surface over the gust front is initiated, triggering new updraft development. Heavier rain, stronger winds and larger hail (than single-cells) can occur, although the chance of tornadoes is still low. Because the storms move as a family, there is also a chance of being impacted multiple times by the strong winds and rain etc. from different cells. They are considerably less potent than supercells, because closely spaced updrafts compete for low-level moisture. 
Image of a family of multicell storms. The storms are developing from right to left, with the wind blowing from left to right, but increasing in speed from the ground up. Cell 4. is the newest, with the updraft just beginning to produce a small cloud. Cell 2. is perhaps the most mature at this stage, with Cell 1. dissipating. Multicell storms are often hard to see from the ground, since their structure is often obscured by low cloud - and they are usually not as well defined as in the picture above. 
From http://www.mykidsway.com/nature-world/thunderstorms.php
Radar reflectivity image of a multicell storm. Often, these storms will appear as a "blob" on the radar at low levels, as precipitation from individual cells merges. At high levels, the individual cells are usually better defined. 
From http://www.weather.gov/gsp/26May06Severe
 
The multicell thunderstorm forms in an environment with speed shear and little directional shear. New cell growth generally occurs on the upstream flank of the storm, where the environmental flow forces air over the gust front, although if the environmental low-level speed shear is strongest, cell growth can be also preferred on the downstream flank of the storm, where the vorticity from the gust front and environmental shear combine constructively to enhance updraft development. As illustrated in panel b) of the diagram below, the cells move rearwards relative to the cold pool, and dissipate once the cold pool undercuts them and prevents further rising motion from the surface. At this stage, a new updraft will develop along the leading edge of the cluster and a new cell will develop, before it too moves rearwards and the cycle repeats itself. The cold pool is constantly enhanced by cold downdraft air from the developing cells, and so has a tendency to move much faster than the storms themselves, undercutting them and giving the image that they are moving rearwards. Each cell lasts 45 min - 1hr from birth to death, although the cluster of storms can persist for several hours. 
 
Importance of vertical wind shear
Single cell storms are associated with very weak shear, resulting in a vertically-stacked structure. The outflow boundary often "outruns" the storm cell and is generally too weak to trigger additional convection, since it is only reinforced by the cold downdraft air from a single cell. Even if new convection develops, it is generally too far away to interact with the parent cell. Speed shear is able to keep the gust front near to the storm updraft by moving the storm with the cold pool. This triggers new convection close enough to the old cells that they can interact each other and form a multicell thunderstorm.
     
A schematic of the different impacts the cold pool from a thunderstorm can have in an environment with no wind shear (a) and an environment with only speed shear (b). The cold pool in (a) has little impact at all, since it is too weak to force new updraft development. However, in (b), new updrafts form on the downwind flank of the initial thunderstorm. 
From Markowski and Richardson textbook: Mesoscale Meteorology in Midlatitudes, 2010). 
However, cells inside the storm cluster do not necessarily move at the same speed or direction as the overall storm system. This is because new cells like to form on the side of the storm where the warm, moist air is located. This is called the preferred flank and is generally on the southern side in the northern hemisphere, but really depends on where the nearest heat and moisture source is. Individual cells always move with the velocity of the mean wind averaged over their depth, and this combined with the repeated development of new cells on the preferred flank leads to a propagation of the storm system that may be slower or faster, and often in a different direction than the mean wind. 
Schematic depicting the individual cell motion versus storm motion. 
From http://www.estofex.org/guide/1_4_2.html