A Calamity is a cataclysmic event that signals the end of an Astral Era, thus leading into a time of hardship known as an Umbral Era. Though there have been seven Calamities total in the history of Eorzea[1], most instances of The Calamity refer to the Seventh Umbral Calamity—the descent of Dalamud in the year 1572 of the Sixth Astal Era, leading to the release of Bahamut at the Battle of Cartenau.
Note that above is a conglomeration of the "The End of an Era" cinematic and cinematics from the Binding Coil of Bahamut, and therefore can be said to contain spoilers.