Lore in Dark Souls 3 will be covered on this page. The game revolves around an "Unkindled Ash" trying to defeat a resurrected Lord of Cinder to restore the linking of the Flame. Throughout the game, you will find Epitaphs that are lore notes engraved into small blue-tinted headstones that will give you bits of lore.
Dark Souls 3 like it's predecessors takes place in the same world and revolves around the first flame, and the cycle of fire and dark. This installment apparently focuses on the "Unkindled Ash" as a dark hero and protagonist that goes to find and bring back 4 Lords of Cinder to their thrones at Firelink Shrine, apparently to link the Flame once again. The Lords of Cinder refer to the guardians of the Flame and they (possibly) linked the Flame along with Gwyn. However, They died and Gwyn was recognised as the only Lord of Cinder. Now the Lords of Cinder have been reborn but, they have left their thrones, and it is up to the Unkindled Ash to restore them to their thrones. There are apparently five Lords of Cinder, Aldritch saint of the Deep, Yhorm the Giant, Farron and his Abyss watchers, Holy King Lothric, and Ludleth the Exiled (Who looks strangely like a small, shriveled, dwarf hollow of Gwyn. This may be why he stil sits on his throne.)
(Further information is speculation based on leaked information and trailers).
It is unclear what age Dark Souls 3 takes place in, either the ending of a new age of fire, or the midst of an age of dark. Similar to Dark Souls 1, it appears that no matter which ending you chose in Dark Souls 2, it would have little affect. Only one definitive ending is choose able in Dark Souls 2, which is to turn your back on renewing the flame in the Scholar of the First Sin DLC. The other ending is unclear on what your character does.
Based on the reveal trailer's opening words, "...And now only embers remain", it makes us think that the fire has almost faded completely and your character is the one chosen to shepard the dark soul that is humanity and break the cycle, or continue the cycle.
Yes indeed it is called Lothric,
Where the transitory lands of the Lords of Cinder converge.
In venturing north the pilgrims discover the truth of the old words.
The fire fades and the lords go without thrones.
When the link of fire is threatened the bell tolls,
Unearthing the old Lords of Cinder from their graves:
Aldrich Saint of the Deep
Farron's Undead Legion the Abyss Watchers
and the reclusive lord of the profaned capital,
Yhorm the Giant
Only in truth the lords will abandon their thrones
and the unkindled will rise
Nameless accursed undead,
Unfit even to be cinder.
And so it is,
That ash seeketh embers.