The Day of Reckoning is an event mentioned in a Datalog within Final Fantasy XIII-2. It is either an alternate name for the time when Etro's gate would be opened, or, more likely, the title given to the moment when the god Bhunivelze would enter the unseen realm and fight his mother Mwynn (for which the great Arks were constructed by the Gran Pulse fal'Cie). The Thirteenth Ark is suggested to play part in the event and act as a shelter for Pulsians during that time.
The event is specifically mentioned in the Datalog for the Coliseum, where it says its combatants will be trapped within the arena until the Day of Reckoning, when they would join the battle between Bhunivelze and Mwynn.
The Day of Reckoning is a common phrase for a final, climatic battle. Specifically, it refers to the Last Judgment, the day when God would judge the world before the final battle of Armageddon.