"The answer is really quite simple. The cells in the superficial or upper layers of skin, known as the epidermis, are constantly replacing themselves. This process of renewal is basically exfoliation (shedding) of the epidermis. But the deeper layers of skin, called the dermis, do not go through this cellular turnover and so do not replace themselves. Thus, foreign bodies, such as tattoo dyes, implanted in the dermis will remain."
James B. Bridenstine of the department of dermatology at the University of Pittsburgh Medical Center adds:
"Our skin is primarily made of the protein collagen, which is produced by cells known as fibroblasts. When the skin (or any other tissue, for that matter) is wounded, the wound-healing process initiates the generation of new fibroblasts to produce scar collagen, which is different from the collagen in normal skin. Even though individual cells within the skin periodically die and are replaced with new cells, the scar collagen remains. The only time when wounds will heal without producing scars is during the fetal stage of life, when the skin produces fetal collagen, a protein that is different from adult collagen. If we could find a way to turn on the production of fetal collagen after birth, then we could, presumably, perform scarless surgery.
"Tattoos remain in the skin because the ink particles that produce the coloration are too large to be ingested by the white bloods cells that patrol the body and carry foreign bodies away from the skin. The new tattoo-removing lasers work because the laser energy pulverizes the ink into microfine dust particles that are small enough to be taken in by the white blood cells and carried away.