Historic Moments are significant events that happen throughout the history of the game, ranging from being the first to choose a new form of government, to building a Unique Unit, to discovering Natural Wonders. There are dozens of these Historic Moments through the course of the game, representing world-first and player first achievements. These moments differ through the eras. For example, clearing a Barbarian Camp in the Ancient Era is an important milestone, but clearing one in the Modern Era won't even merit acknowledgement. You can view your Historic Moments at any time in your game by clicking on the History Timeline in the upper left toolbar.
Each Historic Moment adds towards your Era Score for the era, which is the measure of whether you will rise into a Golden Age or sink into a Dark Age in the next era. If you are gaining lots of Historic Moments in an era, it means you are likely to go into a Golden Age in the next era. If you are the first player in the world to get a Historic Moment, you will earn more Era Score points.