Dear Wayne,
Black spots on the flesh of the potatoes if hard can be cut out and the rest of the potato flesh should be good to eat.  This is usually caused by rough handling.  In other words they are bruised and the flesh turns gray or black in spots.
If the spots are mushy or the flesh around it is mushy, smells musty or any smell that should not be associated with potatoes, discard the entire potato.  This could be mold, or a bacterial infection.
They can also have black spots in the flesh if they are stored in a very cold area for a long period of time.  
If there are any tears in the potato skin, that can cause the flesh to turn black through oxidation. Again, cut the area out and if the rest of the potato is hard, no off smells, it is most likely good to eat.  This is always a personal decision as to whether the potato is consumed or not.
Grayish, or brownish round spots on the skin are referred to as black dot disease, a bacterial infection.  Wash the potato well and cut or scrape off the black spots.  
If you see green areas, trim them off completely before eating.  The flesh is good to eat as long as all green skin and flesh is removed.  When growing potatoes if some of them pop up above the soil, they will turn green from the photosynthesis process.  Cover them with mulch to hide them from the sun.
One other question that is asked is why does the potato have a hole in the center of the potato?  That is called hollow heart.  It is caused by weather changes or any stress that might occur to the plant while developing the potatoes.  If the weather is too cold or cold and hot or too much rain, etc., these are the types of things that cause the hole in the center of the potatoes. 
If you have more questions, please re-contact us.  Thank you for sending your inquiry to us.