Polaris is the north or pole star and the brightest star in the sky, found in the Ursa Minor/Small Dipper constellation. In Norse Mythology the Pole Star is known as Veraldar Nagli or the World Spike, placed at the center of the universe and has the heavens revolve around it.
She is a godlike entity that resides in the Akasha Stratum, and has access to the Akashic Records, allowing her to edit the fate of multiple universes. She has inherited the Heavenly Throne from her predecessors, Kochab, Thuban and Vega, with Er Rai being her direct successor. This implies that Polaris is actually a member of an alien race, rather than a unique being.
Polaris has deemed mankind to be corrupted and irredeemable, and thus used the Void to gradually destroy and remake the world. When the Dragon Stream's barrier over Japan prevents the advance of the Void, she sent the Septentriones to attack and destroy the radio towers projecting these barriers. The Anguished One reveals that once all the Septentriones have been defeated, the path to Polaris will open and allow mankind an audience with Polaris. However, Polaris will only listen to the will of mankind, rather than the wishes of a single person, and thus the Protagonist ends up convincing as many of the Demon Tamers as possible to join the cause he supports.
Polaris is fought at the end of the game, regardless of which route the player picks. What happens afterwards, however, depends on the route. If the player picked Yamato's or Ronaldo's route, she will remake the world according to their ideals. If the player picked Daichi's route, Polaris will either rewind time to before the invasion, or she will be killed by the party, freeing mankind from its cycle of destruction and rebirth. If the player sided with the Anguished One, Polaris is defeated and the Anguished One takes her place on the Heavenly Throne, remaking it and becoming one with the new world.
It's revealed by Tico that on the last day before the world was reset, the Protagonist had recruited all of his friends, including the Anguished One, to free the world from Polaris' control. After Polaris was defeated, the Anguished One replaced her on the Heavenly Throne and reset the world to its previous state, with Polaris ceasing to exist in the new world.
Polaris was first mentioned by Alcor after the fight against Megrez. Alcor reveals to Hibiki that Polaris is the almighty administrator of order and the one responsible behind the whole ordeal. She has arrived at the conclusion that mankind's existence hinders the natural order of the world due to their nature to unconsciously continue to prosper even though they've lost the will to preserve, which is considered as artificial that contradicts the natural order in Polaris' view.
If the player is on the Anguished One's route, the Anguished One must be dispatched, and must survive throughout the entire battle. As the Anguished One cannot overthrow Polaris by himself (or even damage her, for that matter), a game over is also issued if all other human leaders are defeated.
The fight with Polaris comes in 3 stages. During the first stage, Polaris is effectively immune to everything save Almighty (which she resists), being capable of reflecting Physical attacks and nullifying every elemental attack, in addition to possessing several powerful skills and infinite range. Instead, the player has to weaken her by defeating the Guardian Stars scattered across the battlefield, gradually reducing her defenses, before being able to attack Polaris.
Polaris carries a unique skill called "Heaven's Wrath", which deals almighty damage for each Guardian that exists on the field. Therefore, the player must destroy all Guardians in order to minimize the damage. Unfortunately, after several turns, a defeated Guardian will be resurrected. However, Polaris will not recover her skills or resistances that were lost after the defeat of a Guardian.
For the next stage, Polaris gathers all her Guardians to form her full body. What happens depends on the route the player has chosen.
If the player is on the Anguished One's route, the Anguished One is sealed and prevented from taking any actions outside of skirmishes. The player has to attack and defeat Polaris to move on to the next stage, while fending off several strong demon teams that can harm the Anguished One.
If the player is on Yamato's route, the player has to defeat a number of strong demon teams to proceed.
If the player is on Ronaldo's route, fewer demons need to be defeated to proceed. However, Polaris also introduces 2 civilians which can be killed by the demons in one shot, and the players must prevent their deaths. The demons will prioritize on moving towards and killing the civilians.
If the player is on Daichi's route and chose to restore the world, Shadow forms of the Demon Tamers are summoned, mocking the player's choice, and they must be defeated to proceed. If the player has chosen to kill Polaris on that route instead, Polaris decides that their will to fight stems from the protagonist and summons several demon teams to attack the protagonist. Polaris must be attacked and defeated to proceed, but from this point on in the battle, if the protagonist is dead a game over is issued.
For the final stage, Polaris takes on a more bulky form, composed of 3 parts, labelled Polaris A, B, and Ab. Polaris A, the main body, is capable of firing Supernova, a large laser which inflicts Almighty damage to all teams standing in its line of sight. Said line of sight includes the bridge leading up to her. Fortunately, there are 2 conditions to take note regarding Polaris' Supernova:
Polaris always telegraphs the attack by using Star Compression the turn before.
Polaris always rests for a turn after firing Supernova.
When Polaris A uses Star Compression, it is highly recommended to evacuate the two columns leading up to her, unless the team that isn't doing so has resistance to Almighty damage (Yamato, Lucifer, or anyone with Anti-Almighty equipped) or a lot of HP and Vitality to withstand Supernova.
Polaris B, her right shoulder, possesses powerful Phys attacks along with infinite range, and Polaris Ab, her left shoulder, constantly summons another demon team (via Magnetite Conversion) into the battlefield each time it gets a turn. Only Polaris A needs to be defeated to end the battle, but Polaris B and Ab will increase the difficulty in defeating Polaris A by increasing her offensive and defensive strength respectively. Polaris is also capable of reviving Polaris B and Ab if they are defeated.
If the player is on the Anguished One's route or has chosen to kill Polaris on Daichi's route, Polaris B and Ab have their Racial Skills altered to give all Polaris bodies access to Double Extra and a Beast Eye ability, making both of them a greater threat to whoever gets attacked by any part of Polaris. Barring Supernova and Megidolaon, the largest source of damage coming from Polaris and her companion stars is mostly Physical damage, so prepare teams with Physical resistance (preferably Reflect, since Polaris A has Pierce). As Polaris A has no resistances other than an immunity to Curse, you are free to use whatever attacks you see fit to defeat her.
Polaris' predecessors (Thuban and Vega) are stars that acted as the pole star in ancient times. Kawkab, however, means planet in Arabic and is very likely a mistranslation of Kochab, an actual previous pole star. Eli is instead a variant on the name of God as spoken in Arabic, Hebrew, and Aramaic; however, it could likely be a mistranslation of the star Errai, which will indeed be the next pole star after Polaris. In Record Breaker these names are translated correctly.
Polaris survives in the Restorer, Meritorious, and Egalitarian endings as the entire fight with her could simply be a means for her to test humanity's resolve. In the Liberator and Kingmaker endings, however, as Polaris' life is at stake, there would be no need for such a pretense. This would also explain why her "binary stars" have abilities that make the fight much more difficult then, as Polaris is using all her power to oppose the party's intentions.
Polaris, in both Ronaldo's and Yamato's routes says she desires the world to be the way the party wants it to be. For example, she says she also desires a merit driven society in Yamato's route. This gives Polaris a contradictory point of view, but since she appears in the Meritorious ending scene, it is possible that she really wishes for such a world and had lied to Ronaldo, uncovering a darker side of her spirit, or, from a purely holistic point of view, all she wanted was to test the human spirit and ensure the next step in human evolution was taken, no matter in which direction.
Polaris' individual parts of her third form are named after its individual component stars, since the actual Polaris is a multiple star system. Polaris Aa is much larger and brighter than Polaris B and Ab, thus it is used to name the main body. Polaris B is much further from Polaris Aa than Ab, explaining its infinite reach across the battlefield.
If the protagonist chooses to kill Polaris in the Liberator ending, she will tell the party that they will no longer be able to wield the power of demons, making her similar to Babel in a way.
In the game, Polaris only grants the wishes of mankind and will not grant the wish of a single person alone, which means the protagonist must convince as many Demon Tamers as he could to join him. However, in the anime, Polaris will grant a wish for anyone who survives his judgment even if it is only one person.