The Bow was said to have been carried into battle by Auri-El himself against the forces of Lorkhan during the Ehlnofey wars of the Dawn Era. When Trinimac defeated Lorkhan and tore out his heart, Auri-El fastened it to an arrow and shot it long into the sea. The heart landed in the sea and, over the next millennia, Red Mountain eventually formed around it. 
Auriel's Bow has changed hands several times over the years, most commonly appearing in the hands of Tamriel's most famed heroes. The Bow was first found by the Eternal Champion during their quest to overthrow Jagar Tharn during the Third Era. The details as to exact discovery of the Bow by the Etheral Champion remain unclear, although three possible locations relating to where the Champion found the Bow are known (either in Elsweyr, Skyrim or Valenwood).
Six years after the Etheral Champion completed their quest, the Hero of Daggerfall was awarded the Bow by a witch after being given an oppertunity to obtain an Artifact of Power by their Knight Order. The witch (after the Hero had kidnapped her great-granddaughter for her) informed them that the Bow could be found in a far-away dungeon, scattered among some other rusty equipment.
The Bow's location was also offered to the Hero by a mage whom had angered Nocturnal. However, a few moments after the Hero took the Bow from it's location, it fell apart, having been little more than an elaborate fake.Ralyn Othravel. Mistress Therana discovered this fact, having smelt something "like ash yams" (it must be noted that Therana was rather eccentric) in Othravel's vicinity around the Ghostgate. She then sent the Nerevarine on a task to take the Bow from Othravel.  Later, the Museum of Artifacts in Mournhold offered the Nerevarine 15,000 for the legendary weapon.
Once again, the legendary weapon then disappeared, this time for more than two centuries. In what has turned out to be the Bow's most recent appearance, it was rediscovered by the Last Dragonborn in 4E 201 within the Inner Sanctum, hidden deep inside the Forgotten Vale. The Bow was the key to the success of the Volkihar Clan's plan to fulfill the Tyranny of the Sun prophecy, although the Bow was used to kill the Clan's leader Lord Harkon before it could be fulfilled by the Volkihar Clan. 
Auriel's Bow draws its power directly from Aetherius, using the sun as a gateway. The Tyranny of the Sun prophecy had been based on this fact, as the bow drew its power from the sun, it could be used as a tool against it. Despite this direct relationship (which caused the Bow's arrows to appear as if on fire), however, the Bow had also been known to have several other effects, while at other times, had seemingly no effects at all.
The Bow is known to have had many abilities, although these varied between its appearances. Below is a list of all known powers the Bow possessed.
- Never Miss (a unique ability which prevents the Champion from missing their target).
- Sun Damage (tripled against undead)
- Sun Shroud (shrouds the Sun, turning day into darkness - when using Bloodcursed Arrows).
- Sunburst (Bolts of Light rain down from the sun - when using Sunhallowed Arrows).
Interestingly, for the first and so far only time, during the Bow's appearance on Vvardenfell, during the events of The Elder Scrolls III: Morrowind, it did not have any enchantments or effects. The only noticable thing about it was that it "smelt of ash yams".
Auriel's Bow appears in the following games: