The New Republic was the name of the galactic government established by the Alliance to Restore the Republic to replace the Galactic Empire after the Battle of Endor. It was less often referred to as the New Galactic Republic or the Second Galactic Republic. Like its predecessor, it was often referred to as the Republic.
Formed in 5 ABY, after the dissolution of the former Alliance of Free Planets (which was formed directly after the Battle of Endor as an intermediate government), it was to become the supreme governmental authority of the galaxy. The "New" Republic was supposed to be a sort of continuation of the original Galactic Republic that had become known as the Old Republic at that time.
The New Republic was founded on the same premise that the old Galactic Republic was a thousand generations earlier—to have a representative parliamentary body govern the galaxy in a fair and equal manner—and it was designed to eradicate the shortcomings that brought about the Galactic Republic's downfall. The New Republic was protected by the New Jedi Order that was established by Luke Skywalker, just as the Galactic Republic was once protected by the old Jedi Order.