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 Knight Gael and the Firekeeper.
After prince Lothric, the promised Lord of Cinder of the current generation, refused to link the First Flame, the First Flame resurrected four past Lords of Cinder so that they may also relink the flame: Ludleth of Courland, Aldrich, Devourer of Gods, Yhorm the Giant, and the Abyss Watchers.
However, only Ludleth agreed to burn once again, the other three didn't take their thrones and left after seeing the state of the world. The Flame then starts to resurrect a number of Undead, now called Unkindled, in the hopes that they may seek embers and slay the Lords of Cinder and bring their ashes to Firelink Shrine and link the flame. Some of these Unkindled are related to one of the Lords; Anri and Horace were children supposed to act as food to Aldrich but fled, Siegward was an old friend of Yhorm's, and Hawkwood is a former member of the Abyss Watchers.
For that objective, the Unkindled are apparently immune to hollowing, being able to die and revive at bonfires without hollowing, unless they acquire Dark Sigils from other hollows. The Sable Church of Londor revere the Unkindled, as it is prophesied that an Unkindled will become the Lord of Hollows and bring the Age of Dark, but Elfriede ignored this and left to the Painted World for unknown reasons.