These are usually little bits of the lining of the oviduct which got shed during the egg formation process, but some may be partially broken down blood spots. They often take on the appearance of dirt or soil and vary in colour from pale yellow to dark brown or red, as shown in the images. Sizes vary from tiny specks to a few millimetres in diameter. There is a higher incidence of meat spots in eggs from older layers.
Though they look unappetising they are harmless. I usually just pick them out of the egg before cooking it.