Praise the sun
SPOILERS for the Dark Souls games, obviously.
Dragons used to rule the world in darkness
Before humans and monsters and even light entered the world, the everlasting dragons ruled over the land, living in massive stone-like arch-trees that extended so high into the sky that you couldn't see their tops. The dragons were immortal and invincible with stone scales that protected them from any assault. Not much else is known about this time period in Dark Souls' lore, but after a millennia existing as beings of the darkness, the dragons’ rule was ended with the coming of fire.
The light called the First Flame came from the depths of the world and contained extremely powerful Great Souls that give whoever possesses them god-like power. The Souls of Lords were found by Gwyn the Lord of Sunlight, Nito the First of the Dead, and the Witch of Izalith. The Furtive Pygmy, the first ancestor of humanity, gained the Dark Soul. Together they took their newfound power and made war against the ancient dragons.
Light came into the world and the dragons were killed
Gwyn harnessed the power of lightning to destroy the dragons' tough scales, ridding the ancient beasts of their defenses. The Witch of Izalith and her daughters of chaos unleashed great firestorms to destroy the arch-trees. Nito cast a miasma of disease on his enemies and a dragon traitor named Seath the Scaleless betrayed his own kind, sealing the destruction of all the everlasting dragons.
Destroying the dragons brought an age of fire and light and prosperity for humanity. But as time went on, the light of the First Flame began to fade, along with the power of the Souls of Lords. In order to keep the age of fire from plummeting into a new age of darkness, the Witch of Izalith attempted to use her power to create a new flame, but the process morphed her and her daughters into hideous monsters and spread corruption and demons across the land. In a last ditch effort to save the flame, Gwyn sacrificed himself to fuel the First Flame, creating the undead curse and corrupting his army of loyal knights.
The Chosen Undead kills all of the Lords and links (or doesn't link) the Flame
While the Souls of Lords weakened, the Dark Soul, held by the Furtive Pygmy, became more powerful. The Pygmy split the Dark Soul into pieces creating humanity, with each human holding a small portion of the Dark Soul until they lose it and become hollow. With the world corrupted, humans who have lost their humanity and become hollow are sent to the Asylum to await the end of the world. This is where the Chosen Undead (the player) begins his or her journey.
As the Chosen Undead, you travel to the land of Lordran with the goal of linking the fire as Lord Gwyn had, extending the age of fire and undoing the undead curse. To do this, the Chosen Undead travels to the city of Anor Londo to gain possession of an artifact called the Lordvessel. Once obtained, you must gather the Souls of Lords held by the corrupted Witch of Izalith, Seath the Scaleless, Nito First of the Dead, and the Four Kings of New Londo who possess the shards of Gwyn's Soul of Lords.
The Chosen Undead faces the Lord Gwyn and makes a choice
Once you defeat the last of the old gods, the Chosen Undead meets a primordial serpent named Darkstalker Kaathe. Kaathe reveals that the Furtive Pygmy, the first human, was the owner of the Dark Soul and planned to wait for the Age of Fire, the age ruled by the gods, to wither and fade. When this happened, the humans could usher in an age of dark, but the rise of humans was delayed by Gwyn, altering the natural course and cycle of the world.
From this point, the Chosen Undead faces Gwyn, and upon his defeat is given a choice: follow in Gwyn's footsteps and link the fire, prolonging the Age of Fire, or usher in a new Age of Dark, becoming the Dark Lord of that time. The course you choose doesn’t change the events of Dark Souls 2 because, well, the world is destined to continue the cycle of light overcoming dark then back again.
The Flame reignites and the Kingdom of Drangleic is formed
Hundreds of years after the events of the first Dark Souls, the kingdom of Drangleic was founded by King Vendrick on the bones of the land's former inhabitants. Vendrick and his brother Aldia conquered the kingdom with Dragon Riders and countless warriors. But after years of prosperous rule, Drangleic was overcome with the curse of the undead.
King Vendrick did everything he could to stop the plague from spreading, sending his accursed subjects to faraway lands and locking them up in a prison called the Lost Bastille. He also sent his knight, The Pursuer, to destroy every undead creature. But the king's obsession with ridding himself and the kingdom of the curse didn't end there, and his fear led to some misguided decisions and disastrous events.
King Vendrick wages war on the Giants
King Vendrick's queen Nashandra convinced the ruler that he needed the power of great souls to stave off the undead curse, so he crossed the nearby sea and attacked the land of the Giants (owners of particularly powerful Giant's souls). Vendrick's forces were driven back to Drangleic and his kingdom was invaded by the massive Giants for generations until the Giants were eventually defeated.
At some point, the King discovered that his queen Nashandra wanted access to the Throne of Want, which housed the First Flame and a Great Soul, so he fled to Drangleic's Undead Crypt to keep his King's ring (the key to the throne) from her. Unable to pursue her goals herself, Nashandra sends the Chosen Undead (who is tricked into doing her dirty work) to unlock the path to the Throne of Want. But after discovering her true intent, the Chosen Undead confronts and defeats Nashandra at the Throne and is once again able to choose whether to link the fire or leave the First Flame to be absorbed by the dark.
Vendrick's brother Aldia is a mad scientist
While Vendrick waged war on the Giants, his brother Aldia worked in his secluded manor to attempt to discover the source of the undead curse through experimentation on various creatures. He may have been responsible for creating the wyrms that the King's Dragon Riders rode into battle.
Aldia, in his experiments, created the Emerald Herald, your guide and stats leveler throughout the game. It was also theorized that Aldia may have (according to fan speculation) attempted to change himself into the form of an ancient dragon but it turns out he messed up and changed into a monstrosity instead.
The Souls of Lords are carried in new vessels
What connects the stories of the two existing Dark Souls games is the continuing theme of cyclical reincarnation. In the age of Dark Souls, the Flame has been reignited and the ancient souls have been reborn in new bodies. The Souls of Lords still exist, but they are held by new beings that give nods to the Lords of the original Dark Souls.
The Dark Souls 2 bosses The Rotten, Old Iron King, Lost Sinner, and the Duke’s Dear Freja all drop great souls that coincide with Dark Souls bosses Nito, Gwyn, the Witch of Izalith, and Seath the Scaleless, respectfully. Now, with Dark Souls 3 just on the horizon, it will be interesting to see how the game integrates the old souls into some new, monstrous bosses.
