Skip the next bit if you want to avoid potential spoilers, but here's a very brief overview: Viserys was a well-loved king, and House Targaryen — and Westeros — thrived under his rule. Still, while he was alive the question of who would be his heir created factions in his court. Viserys had named Rhaenyra, his daughter from his first wife, his heir, but his second wife, Alicent, believed her son Aegon should be next in line to the throne. Meanwhile Viserys' brother, Daemon, also wanted to stake his claim.
When Viserys died in his sleep, Alicent's eldest son was crowned Aegon II, causing the outbreak of civil war between him and Rhaenyra which became known as the Dance of the Dragons.
It's unclear at what point in the story House of the Dragon will pick up, but it seems likely it will be the latter years of Viserys' reign, with the two factions already at odds with each other. And, of course, House of the Dragon will have all the hallmarks people recognise from Game of Thrones: political intrigue and schemes, complicated and fraught family dynamics, love and betrayal, beautiful dragons, epic battles, lots of blonde wigs, and, er, the odd bit of incest.