"Auriel's Bow appears as a modest Elven Longbow, but it is one of the mightiest weapons ever to exist in Tamriel's history. Allegedly created and used like its sister, Auriel's Shield, by the great Elvish demi-god, the Bow can turn any arrow into a missile of death and any wielder invulnerable to any lesser attacks. Without Auriel's power behind it, however, the bow uses its own store of energy for its power. Once exhausted of this energy, the bow will vanish and reappear wherever chance puts it. Its most recent appearances have been subject of gossip for hundreds of years."
After joining any Knightly Order and having a high amount of reputation, this quest can become available.
Accept the offer to receive an "ancient artifact of power" from a member of the order. Go to the dungeon mentioned by the employer and speak to the witch. She will ask the Agent of Daggerfall to kidnap her great granddaughter and, upon returning to her, she will then guide them to a dungeon containing either the bow, Auriel's Shield or the Lord's Mail.
Upon progressing through Nocturnal's quest and confronting the Battlemage, there is an alternative to accept the mage's offer and retrieve Auriel's Bow from a location he gives. If the bow is already obtained it cannot be received again, so instead he will offer the Warlock's Ring.