The Omega Weapon can be found in the Omega Ruins, at the very end of the dungeon. This means that you will first have to defeat the Ultima Weapon.
In many ways, Omega Weapon is an upgraded version of Ultima Weapon. It is able to inflict Confuse and Petrify on a character, so having Stoneproof on your armor will keep your party members from being shattered. It is also immune to all negative status effects, and absorbs all elements.
Omega Weapon has four attack patterns that it cycles through. For each stage, it will use either a physical attack (which deals around 4000 damage) or an ability. Using the ability will cause it to cycle to the next pattern.
First pattern: it could use Core Energy, which will deal between 4500 and 9999 damage to one character.
Second pattern: it has a chance to use either Confuse or Break.
Third pattern: it could use Shimmering Rain, which deals 2500 damage to the party.
Fourth pattern: it could use either Demi or Ultima.
When an Aeon is out, the boss will only cycle between the Core Energy and Shimmering Rain stages. Omega Weapon also has a hidden counter throughout the fight, which increases by 2 when it uses an ability, or 5 when using a physical attack. Once this counter hits 30, it will cast Nova, which deals around 7000 damage to the party.
Haste, Protect, and Shell are important to keep up in order to keep your damage up while reducing the damage you take. Avoid using elemental magic and weapons, as you will only heal Omega Weapon. It has 999,999 HP, so this will be a long fight. If you get Celestial Weapons beforehand, pairing them with Quick Hit will greatly speed up the fight.
When defeating Omega Weapon, you will earn 50,000 AP (60,000 on Overkill), and 20,000 gil.
Omega Weapon is one of two bosses where Kimahri can learn Nova from by using Lancet on it.