Unkindled refers to beings who are cursed with undeath, and branded with the darksign. It is unclear what caused the change in terms from Undead, however given that hollows have also visibly changed in Dark Souls III, it can be assumed that there is some distinction between them and the older "Undead".
Those who are Unkindled appear to be a sort of opposite to the Undead seen in other games. Whereas the Undead seek Humanities or Human Effigies, objects comprised of some form of darkness, Unkindled seek Embers, small remnants of flame. From this, we can infer that whereas Undead were lacking Dark in previous games, in Dark Souls III they must lack flame. Oddly, certain characters seem to be reverent of the Unkindled, such as Slave Knights Gael, and the Firekeeper.