In the end it took one man, one Jedi to bring about the end of the Sith. That man was a young Jedi by the name of Luke Skywalker. Part of the Alliance and the Rebellion against the Empire, Luke was, for all intents and purposes, the last remaining Jedi. Within Skywalker rested the galaxy's last hopes of salvation. Singlehandedly and alone, Luke confronted Darth Vader and the Emperor at the height of the Rebellion over the forest moon of Endor. During the showdown, Skywalker not only faced the two reigning Sith Lords, but his father as well. The man who would become Darth Vader was also Luke's father. In the end, Luke's strength as a Jedi Knight not only destroyed the Sith, but saved his father as well.
While the galaxy felt the aftershocks of the Sith's presence throughout the galaxy for many years after, the Sith truly died at Endor. The two biggest Sith threats to face the galaxy after Darth Vader and the Emperor's death came in the form of an ancient Sith Lord and the Emperor's secret experiments during his reign. While alive, the Emperor experimented with cloning as a way to prolong, and, if capable, escape death. With the Empire left in shambles, a power grab was made by its many leaders, as well as the Dark Jedi the Emperor tasked to safeguard his clones. Luke Skywalker and the Alliance spent the better part of a year fighting these Dark Jedi and Emperor clones.
Also, as Luke Skywalker worked to rebuild the Jedi Order and established his Jedi Academy on Yavin IV, an ancient villian reemerged to again plague the Jedi and make a grasp at immortality. Within the ancient Massassi ruins of Yavin IV, the spirit of Exar Kun was awakened, the Dark Lord of the Sith who was imprisoned within the temples four thousand years prior during the Sith War. It took the strength of Luke and all of his Jedi trainees to finally smother the ember of Exar Kun's spirit left smoldering for four millenia.
Finally, as there always had been and always would be throughout history, there were those who were enamored with the Sith legends, and seeking power themselves, would attempt to resurrect ancient Sith traditions and create new Sith brotherhoods. The Disciples of Ragnos were one such group whose ultimate goal was to resurrect the spirit of Marka Ragnos. Also, there was Lady Lumiya, an apprentice of Darth Vader's during the height of the Empire. After its defeat, Lumiya made her grab for power and confronted Luke Skywalker on multiple occasions. In reality, these Sith threats were a mere pest to the Jedi and the New Republic. These Force-sects paled in comparison to the might and the glory of the ancient Sith they held in such high regard. The true Sith line was ended at Endor with the deaths of Darth Vader and the Emperor at the hands of Luke Skywalker.