While some tattoos can be temporary, shedding off when the skin is sloughed off, permanent tattoos remain because
1) the ink is injected within the stratum granulosum, where the ink granulates within the skin, staining it permanently.
2) the ink is injected within the stratum corneum, the most superficial layer of skin. injecting ink any deeper than this would not be seen.
3) the ink is injected into the hypodermis, the deepest layer of the integumentary system. the deeper the tattoo is, the more permanent it will be.
4) the ink is injected into the dermis, the layer just below the epidermis, which has a direct source of nourishment from capillaries.