Kentucky's oldest rocks crop out in a wide, gentle uplift in the north called the Jessamine Dome, a particularly high part of the Cincinnati Arch. Younger rocks, including thick deposits of coal laid down during later periods, have been eroded away, but Silurian and Devonian rocks (lilac) persist around the edges of the dome.
The coal measures of the American Midwest are so thick that the rocks known as the Carboniferous Series elsewhere in the world are subdivided by American geologists into the Mississippian (blue) and Pennsylvanian (dun and gray). In Kentucky, these coal-bearing rocks are thickest in the gentle downwarps of the Appalachian Basin on the east and the Illinois Basin on the west.
Younger sediments (yellow and green), starting from the late Cretaceous, occupy the Mississippi River valley and the banks of the Ohio River along the northwestern border. The west end of Kentucky is in the New Madrid seismic zone and has a significant earthquake hazard.
The Kentucky Geological Survey Web site has much more detail, including a simplified, clickable version of the state geologic map.
More Kentucky resources on About.com:
Kentucky Geography, State Symbols & Facts
Kentucky National Parks
Kentucky State Parks
Kentucky Military History Sites
Kentucky Thoroughbred Farms