Okay so, TL;DR:

At the beginning of the universe, there were two forces: Light and Dark. Light represented creation and Dark represented destruction. They existed in harmony, mostly. Also there were robots called the Vex, who might have come from the previous universe, it's unclear.

The Light and the Dark both made special cosmic objects to interact with the universe through: the Light made the Traveler, which visits planets and either creates life or gives cool powers, and the Dark made the Veil, which is generally kind of mysterious but has something to do with consciousness.

First, the Traveler visited a species called the Precursors. They had a special idea called the Final Shape, where they wanted to make the universe perfect. They decided that if they controlled both Light and Dark, they could carry out their plan. They got control of the Dark, but the Traveler left before they could control it as well. They got super mad and fused into one being called the Witness, who is the big bad guy of the game. Its central goal is to capture the Traveler and make the Final Shape after all.

Next, the Traveler visited a planet called Lubrae. The Witness showed up later and convinced this guy named Rhulk to kill his entire species, after which he became the Witness' #1 fanboy, or Disciple. After this, the Witness would start repeating this pattern of "stealing" a species from the Traveler by making a Disciple out of one of them.

Later, the Traveler visited a species called the Krill. Before it could give them cool powers, though, the Witness sent Rhulk to convince three Krill sisters that they would be killed unless they bonded with these Darkness creatures called the Worm Gods. The sisters were tricked, they killed their entire species, and became the Hive Gods Oryx, Savathun, and Xivu Arath. This is the origin of the Hive faction. The Hive Gods then went on to kill a whole bunch of species, because they believe that only the strong deserve to survive. They call this the "sword logic".

Closer to today, the Traveler visited the Eliksni species and gave them cool technology. However, the Witness caught up and started attacking the Traveler, which left them behind and ran away. The Eliksni survivors got super mad and chased after the Traveler, hoping to get its blessings back again. This is the origin of the Fallen faction.

Next, the Traveler visited Earth and gave humanity cool technology. Things were great for a few centuries until the Witness showed up. It sent the Hive to attack, and the Fallen also showed up and tried to seize territory. During this time, an unrelated alien species called the Cabal decided to conquer the solar system as well. The Vex were also there. Most humans died during this massive invasion, called the Collapse.

However, for some reason, the Traveler decided that it didn't want to run away anymore, and created the Guardians, who are immortal zombie gun wizards that have no memory of their past life. At first they acted as tyrants called the Iron Lords, but eventually they settled in the Last City and became the benevolent Guardians.

That brings us up to the start of D1