When George Lucas returned to making Star Wars movies, he chose to go back in time to tell the story of Anakin Skywalker before the events of the original trilogy, but that didn't mean fans were deprived of further adventures of Han, Leia, Luke, and other Star Wars favorites from the original trilogy. The former Expanded Universe stretched several decades beyond the Battle of Endor, telling the story of the classic heroes, their children, and even a number of brand new characters not featured in the movies.
While George Lucas himself never actually considered the Expanded Universe as canon, that stance was made official after Lucasfilm was sold to Disney and those stories were all designated as the alternate universe of Star Wars Legends to make way for a brand new shared universe, only incorporating the Lucas produced movies and The Clone Wars animated series as its foundational canon.
This chronicling is fairly high level and won't cover every individual character arc, but will detail important events that shaped the galactic military and political landscape between the two trilogies.
Page 1 (this page): Palpatine's Contingency and The New Republic
Emperor Sheev Palpatine may have been overconfident, which was his ultimate undoing, but he wasn't as blind to Luke Skywalker's potential to defeat him as he may have appeared. Years before Luke Skywalker was even born, before Anakin Skywalker became Darth Vader, the Sith Lord recruited a young slave boy from Jakku named Gallius Rax, charging him with a massive project. Rax was left on Jakku to oversee a massive mining project drilling to the planet's core, while a local outpost (with some help from Grand Admiral Thrawn) charted navigational routes through treacherous uncharted space into the Unknown Regions where Palpatine could sens a fount of dark power.
By the time of the Battle of Endor, most of Palpatine's preparations were in place, so he tasked Rax with taxing a sizeable portion of the Imperial Navy into hiding, set to trigger the first stage, "Operation Cinder," should the Emperor fall during the battle. Since that's exactly what happened, Rax commenced Operation Cinder, triggering a massive planetary attack against locations of importance to Sheev Palpatine, such as his home planet of Naboo. It appears his intent was to wipe certain details about his life and operations away, although the true correlation between all the targets of Operation Cinder isn't yet clear. Fortunately, the Rebels, AKA the Alliance to Restore the Republic, was able to prevent total catastrophe in many instances, including Naboo.
Formation of the New Republic
Formed under the leadership of former New Republic senators like Mon Mothma and Bail Organa (and even his adopted daughter, Leia Organa), the intent of the Rebel Alliance was always to return the galaxy to the democratic rule of a republic, as it was before the rise of Emperor Sheev Palpatine.
So, naturally, after the Empire's defeat at Endor, the first order of business was to establish the New Republic as the central governing authority in the galaxy. Unfortunately, not everyone was quick to join. Some systems were still under the control of the fragmented Empire, some didn't have faith in the new government that quickly rose out of the Republic, and others were opposed to any strong central government, having opposed the Empire to escape any sort of non-local authority in the first place.
The initial home of the Republic was Mon Mothma's homeworld of Chandrilla, but the capital soon started rotating to new planets on a regular basis to prevent any one location from becoming too entrenched politically as Coruscant had become, which ultimately played a role in Palpatine's rise to power. Slowly, more systems came around to join, especially after the (powerless, but symbolic) Imperial figurehead Mas Amedda publicly surrendered after the Battle of Jakku. The Galactic Concordance was signed, signaling the demilitarization of remaining Imperial forces in total surrender to the New Republic.