Tattoos are permanent because the ink is injected into the dermis layer of the skin, which is composed of collagen fibers, glands, blood vessels, and nerves. The ink is deposited deep enough to avoid being shed with the outermost layer of the skin, the epidermis, which gets shed and ends up as dust mostly. The immune system plays a role in making tattoos permanent. Every time the tattoo needle makes a hole, macrophage cells go towards the wound to try to close it up. The macrophage cells gobble up the ink, which is a foreign invader, to try to get rid of it. But instead, those macrophage cells with bellies full of ink get stuck in the gel-like matrix of the dermis, and the ink travels from one generation of cells to the next, making the tattoo permanent.
Tattoos do naturally fade over time as macrophages continue to carry some of the pigment off, and UV light can also break down the pigment of the ink. However, the ink remains in place until the dermal cells die, and when they have completed their life span, they are absorbed by younger cells, which means that the tattoo remains in place.
In summary, tattoos are permanent because the ink is injected into the dermis layer of the skin, which is composed of cells that do not shed and are absorbed by younger cells when they die. The immune system plays a role in making tattoos permanent by trapping the ink in the dermis layer of the skin.