Tattoo ink stays in the skin forever because it is stuck in the dermis thanks to cells called fibroblasts, according to the Atlantic. Although most of the ink is trapped in fibroblasts, some of it is stored in macrophages, special cells sent out to destroy foreign objects like tattoo ink.
When a tattoo needle enters the skin, it creates wounds. These wounds cause the skin to swell and prompt the macrophages to activate by "eating" the ink. As the swelling goes down, much of the ink that was eaten by the macrophages stays within the dermis, while the rest lies within the fibroblasts. The ink injected into the skin is dark, and therefore shows through the epidermis, according to the Atlantic.