The New Republic is the government set up by the Rebel Alliance following their capture of Chandrila from the Galactic Empire. The New Republic was run by Rebel Alliance leaders Mon Mothma and Leia Organa, and was in a constant state of strife during it's first decade or so in rule. The New Republic lasted for three decades, before being voluntarily dissolved by the First Order, which helped begin the Resistance.
In games (Legends)
The New Republic has appeared very sparingly in games, with only the Jedi Knight series of games wishing to venture into the Expanded Universe and the events after the Battle of Endor. It is likely that the time period has been avoided by Lucasarts because they believe that it would only appeal to hardcore Star Wars fans, and that casual fans do not want to see what happens after the movies. Technically, Star Wars Battlefront: Renegade Squadron is framed from the time period of the New Republic, by having Col Serra narrate his experiences long after the war has ended, but all the events shown in the game take place during the Galactic Civil War