|From It's Okay to be Smart|
First, you stick a needle in your arm and that causes a wound. So your body goes "oh crap! Wound! Get the immune system cells to the site! We will inflame the area with macrophages (cells that eat the "invading material", or the ink, to get rid of the resulting inflammation from your wound)."
Then, some of those cells carry dye away with them to the lymphatic system, but others keep hanging out in your skin (aka dermis), and the ink they're holding onto stays visible because it's right there in the skin!
HOWEVER, your skin slowly heals, replacing the skin cells holding the dye with new, dye-less cells. This makes the ones that are in the "gel-like matrix of the dermis" fade and are replaced. Not all of the cells do this though! Dermal cells are taken up by younger cells when they die, but the new cells take in the ink as well. So that ink stays put. Which is why tattoos get all faded and ugh-like after a while and you have to touch them up. They are permanent - sort of.