The Anvil is precip free and often cold, there is very little moisture in the anvil. clouds take their anvil shape when the air that rises reaches 40,000 or more feet.
The Wall Cloud forms near the downdraft/updraft interface. the interface is the area between the precip area and precipitation free base. wall clouds form when cooled air from the downdraft is pulled into the updraft. Wall clouds are very common amoung supercells but little only produce tornadoes. Wall clouds that are present for 10 minutes or more, move violently up or down mean that a tornado is very likely.
Precipitation free base is obviously by the name is precip free and is located at the bottom of the main updraft and is the main area of inflow. observer may not be able to see precipitation falling but large hail and and rain may be falling from this area.
Precipitation area is the area with the most precipitation or the heaviest rain.