What can the color of feces tell you?

Can tell a good deal. Black stool can indicate internal bleeding. Red stool also can be blood. Clay colored stool can indicate liver disease. Mostly, color is a marker of your diet.
A lot, actually. The usual brown color of stool is a result of your body breaking down the hemoglobin lost by dying red blood cells. If, for example, a patient were in liver failure, this breakdown would not occur and the stool would be much more pale in color. There are many other conditions which can change stool color. Not to mention, the body may struggle to absorb food colorings and dyes (think kids' cereal).
Bleeding, other. Stool is generally shades of brown; from bile as well as food effects. Black stool is seen with GI bleeding, or charcoal pills, or iron, bismuth, some antacids, arsenic. Dark stool, not black, can be from blueberries, greens, etc. Maroon stool can be from bleeding, beets, red foods. Pale: blockage of bile; silver/grey: blockage of bile w/blood. White: barium, some antacid. Yellow/greasy: fat.