Its called a Gust Front - usually visible on radar. When a large thunderstorm cell or complex reaches its peak and begins to dissipate, all of that upflowing air that built up the storm begins to descend at an ever increasing rate as the storm weakens. When it hits the ground it fans out in a gust front. When the gust front passes over your location the temperature will often drop sharply and the winds can get quite gusty - sometimes gusting to 40mph. The temperature differential forms the low clouds - and sometimes the counterclockwise rotation can create a roll cloud. Because of the increased force from the downdraft of descending air, the gust front moves out ahead of the dying thunderstorms at a faster rate than the movement of the thunderstorm itself.
Cheers,
ED