Game of Thrones begins roughly 16 years after the end of Robert's Rebellion, but while it detailed those events it didn't show too much of what happened in between. Robert's Rebellion is one of the most important events in recent Westeros history, including the fall of the Targaryen dynasty and the rise of a new royal lineage.
Much of what happened, or at least the biggest events, was at least covered on Game of Thrones, with some shown in flashbacks. We saw the infamous Tower of Joy scene, where Lyanna Stark gave birth to Jon Snow but died doing so, and saw a brief vision of Jaime Lannister killing the the Mad King Aerys II Targaren, all of which came towards the very end of Robert's Rebellion. Earlier in the timeline, Game of Thrones season 7 revealed the secret marriage between Lyanna and Rhaegar.
Robert's Rebellion may have been built on a lie, thinking Lyanna had been kidnapped, but it was nonetheless successful. The Targaryens were defeated, Robert took the crown, and then a decade-and-a-half later Game of Thrones begins with the death of Jon Arryn, and Robert going to offer Ned the job of Hand of the King. There's a big gap in between all of that, so what exactly happened?
Assault On Dragonstone & The Birth Of Daenerys Targaryen
In the immediate aftermath of Robert's Rebellion, the Targaryens weren't completely defeated, despite the deaths of Aerys, Rhaegar, and his children, Rhaenys and Aegon. Queen Rhaella Targaryen, the sister-wife of Aerys, had been sent to Dragonstone with her second son, Viserys, after Rhaegar's death, and the castle served as the last Targaryen stronghold in Westeros. Seeking to eradicate the Targaryens for good, Robert had his brother, Stannis, build a new fleet and sail to take Dragonstone from the Targaryens.
Before Stannis and his fleet arrived, a great storm raged around Dragonstone, during which Rhaella gave birth to Daenerys Targaryen (hence the name Stormborn). Rhaella died, and some members of the garrison on Dragonstone were prepared to sell Viserys and Dany to the Baratheons in order to survive. Ser Willem Darry was able to smuggle the children away from Dragonstone under the cover of night, taking them to Braavos, where they lived with him in the fabled house with the red door (which didn't make it into Game of Thrones). Stannis, aboard his new ship Fury, unsurprisingly conquered Dragonstone and defeated the last of the Targaryen loyalists, and was rewarded - much to his anger - with being named Lord of Dragonstone, while his younger brother Renly was given Storm's End.
The Reign Of King Robert I
The Robert Baratheon we meet at the beginning of Game of Thrones is an overweight drunk, which is a far cry from the one who led the rebellion. Once described as being "muscled like a maiden's fantasy", being a King didn't exactly suit Robert, although the realm was generally peaceful during his reign, ruling with Jon Arryn as his loyal Hand. He married Cersei Lannister, restoring pride and power to the house after the Mad King had turned against his former friend.
Robert's reign was mostly considered a prosperous one for Westeros, although much of that came because he borrowed so much money, building huge debts with both the Iron Bank and Tywin Lannister. Nonetheless, it was a big improvement over the previous Targaryen rule: Robert cared little for the politics of ruling, but at least he wasn't cruel like Aerys. His marriage did sour though, with Robert fathering a number of bastards, and there were those who called him Usurper. Dorne, in particular, didn't forgive Robert or the Lannisters for what happened to Elia Martell and her children, and Oberyn actually plotted a rebellion, even travelling to Braavos to meet with Ser Willem Darry and make a marriage pact between Viserys and Arianne Martell (one of many book characters not in Game of Thrones), before Jon Arryn travelled to Dorne to meet with Doran Mortall and cool talk of a rebellion.
The Greyjoy Rebellion
Just because the Dornish rebellion didn't happen doesn't mean there was no uprising against King Robert Baratheon, with the Greyjoy Rebellion coming around six years into his reign. After the death of his father near the end of Robert's Rebellion, Balon Greyjoy assumed control of the Iron Islands and wanted to return things to the Old Way. He spent years building the Iron Fleet, and eventually proclaimed himself King of the Iron Islands and declared open rebellion on the Iron Throne, seeking independence from the Seven Kingdoms.
The Greyjoy Rebellion got off to a winning start with a victory at Lannisport, where the Greyjoys burned the Lannister fleet in a surprise attack planned by Euron and carried out by Victarion. However, he lost one of his sons during an attack on Seaguard, before the Iron Throne struck back in full, culminating at the Siege of Pyke (which was mentioned a couple of times during Game of Thrones). Led by Robert and Ned, finally reunited after the end of Robert's Rebellion, the crown smashed Pyke: Thoros of Myr, flaming sword in hand, was famously first through the breach. Jorah Mormont was close behind, and received a knighthood for his efforts. The Siege of Pyke was the end of the Greyjoy Rebellion, and resulted in Balon Greyjoy having to bend the knee and swear loyalty to the Iron Throne and House Baratheon. His eldest remaining son, Theon, was taking into the care of Ned as a hostage to ensure his good behavior.
Mance Rayder Becomes King-Beyond-The-Wall
Mance Rayder didn't appear in Game of Thrones until season 3, but he was already active long then. The exact timeline of Mance's departure from the Night's Watch and subsequent rise to King-Beyond-The-Wall is unclear, but we know that it must've happened after Robert's Rebellion, as Mance visited Winterfell when Jon and Robb were young. He was attacked while out ranging and healed by a wildling woman, who mended his cloak with red fabric. Upon returning to the Night's Watch, he was told to change it to black, which instigated his decision to (quite literally, in this case) turncloak against it. He left to live among the wildlings, and later spent years bringing together various tribes into one, beating five others to become the true King-Beyond-The-Wall.
Main Game Of Thrones Characters Are Born & Summer Begins
Of course, in terms of how it relates to the events of Game of Thrones itself, then the period between Robert's Rebellion and Game of Thrones is when the majority of main characters - and especially the members of House Stark - were born. Jon Snow was born just the end of the rebellion, and was already at Winterfell by the time Catelyn returned from Riverrun with a baby Robb. Sansa, Arya, Bran, and Rickon all followed in the intervening years. It was also here that spring ended and summer was declared, officially starting in 289 and lasting for 10 years, the longest in living memory. Of course, this was starting to come to an end at the beginning of Game of Thrones, which almost immediately promised that "Winter is coming".