Although it is about a Targaryen Civil War, House of the Dragon isn't the story of the Blackfyre Rebellion, nor does it feature the Mad King Aerys II Targaryen. When House of the Dragon was first announced, its story wasn't fully clear, since it was simply billed as an adaptation of George R.R. Martin's book Fire & Blood, telling the Targaryen dynasty's story that begins 300 years before Game of Thrones. That's a rich tapestry, of course, and far more than any one show could fit in, covering the rise and fall of the dynasty with countless wars in between, because the history of House Targaryen is an incredibly detailed one.
Two of the biggest Targaryen stories are those involving the Blackfyre Rebellion and the Mad King. The Blackfyre Rebellion —the first of them, as there were five in total — was a war of succession fought after King Aegon IV Targaryen legitmized his bastards while on his death bed. Meanwhile, the Mad King was, of course, the last Targaryen King, who became increasingly cruel and unstable. As told in Game of Thrones, the Mad King was killed by Jaime Lannister at the end of Robert's Rebellion, with Robert Baratheon replacing him as king.
Both of those are stories worthy of telling, but neither is what the upcoming spinoff is about — although it's easy to see why there may be some confusion, especially for those less familiar watching the House of the Dragon trailer. The Blackfyre Rebellion is a Targaryen civil war led by the great, roguish warrior Daemon Blackfyre, while House of the Dragon features a Targaryen civil war in part led by the great, roguish warrior Daemon Targaryen (Matt Smith). Meanwhile, there's a Targaryen king on the Iron Throne in House of the Dragon, but it's Viserys (Paddy Considine), not Aerys. This is a very different story, far removed from both of those tales.
House Of The Dragon's Time Period: How Long After Aegon The Conqueror & Before Game Of Thrones
Instead of the Blackfyre Rebellions or Mad King, House of the Dragon's time period is 101 years after Aegon's Conquest. Daemon Targaryen was born in 81 AC, whereas Daemon Blackfyre wasn't born until 170 AC, 40 years after the former's death. The First Blackfyre Rebellion begins in 196 AC, with various more civil wars fought over the decades. The final Blackfyre Rebellion, the War of the Ninepenny Kings, took place in 260 AC, shortly before the would-be Mad King took the Iron Throne. Aerys was born in 244 AC, becoming King in 262 AC, so is similarly far removed from the story of House of the Dragon.
The Mad King's story is closely linked to Game of Thrones, directly inspiring its events, but the same cannot be said for the prequel. House of the Dragon begins just under 200 years before Game of Thrones and ends around 170 years before it. Still, House of the Dragon's story will have an impact, since it leads to the end of the dragons, and House Targaryen is never the same after the civil war.