If you've even watched, read, or even vaguely smelled something tangentially related to Game of Thrones, then you already know who the Targaryens are. Daenerys Targaryen and her bleach-blonde ancestors are the fiery end of the Song of Ice and Fire: they're all about dragons, incest, and conquering folks with fire and blood. They forcefully united the Seven Kingdoms of Westeros all those years ago, and forged the Iron Throne. They ruled as kings and queens for centuries. And, according to Dany and her brother, they are still the rightful heirs to the western continent.
So why are there so few of them left?
Ever since Dany's brother was "crowned" with molten gold, she's been the only official surviving Targaryen. There's also Jon Snow, who is almost certainly a secret Targaryen love child. And in the books there's a kid who claims to be baby Aegon, the supposedly dead son of Prince Rhaegar Targaryen.
But that's it. All the other Targaryens are dead. Which seems kind of... strange? Real life royal families have a whole mess of cousins and uncles and whatnot who can lay claim to the throne, but by the time Dany was born, her family was really just down to her parents, her siblings, and her brother's two kids.
And that's because of that one time that a whole generation of Targaryens basically died in a fire.
Yes, despite the fact that TV Daenerys appears to be made entirely out of asbestos, the Targaryens are not actually immune to fire. They can withstand slightly hotter bubble baths, but that's about it. Dany's miraculous dragon-hatching bonfire was a one-time, blood magic deal. She's called the "Unburnt" because she survived fire once, not because she's magically fireproof forever. This seems to be a place where the HBO show has just straight up diverged from the books.
And I guess it's fine if the television writers want to make Dany a super special fire mutant. But it's kind of important that the rest of her family can be burned, because a whole bunch of them canonically burned up in the Tragedy at Summerhall.
Back before Dany was born, you see, the Targaryens liked to get out of the city every once in a while and kick it at their weekend getaway, Summerhall. During the reign of Aegon V, the king summoned most of his family to the pleasure palace to celebrate the birth of his first great-grandchild, Dany's older brother Rhaegar. While he had everybody there, King Aegon decided that it would also be chill to fiddle with some dark magic fire rituals.
This King Aegon used to be little Egg, the squire from Martin's prequel novellas about Ser Duncan the Tall. He's also brother to Maester Aemon from A Song of Ice and Fire, one of the few old school Targaryens left alive (for a couple books, at least). Egg grew up to be a pretty cool king, but as he got older, he started to obsess over dragons. The Targaryen dragons went extinct years ago, but there were still a few dormant dragon eggs lying around. Egg was hoping to use magic to hatch one of these eggs.
As we know from Dany's own experience with dragon eggs, though, they need fire to hatch. They also maybe need human sacrifices. When Dany built Drogo's funeral pyre, she burned the witch Mirri Maz Duur alive before walking into the flames herself. Without realizing it, it seems like she stumbled on the exact combo of magic needed to wake those baby dragons up.
With King Aegon, it's unclear exactly what he tried... but he definitely used fire. In The World of Ice and Fire, Martin includes this damaged fragment from Archmaester Gyldayn, describing what went down during Aegon's experiment:
"the blood of the dragon gathered in one …… seven eggs, to honor the seven gods, though the king's own septon had warned …… pyromancers …… wild fire …… flames grew out of control … towering … burned so hot that …… died, but for the valor of the Lord Comman ..."
Aegon's egg-hatching ritual went horribly wrong, killing him and nearly all the other Targaryens present, as well as their spouses and friends. The castle was burned to the ground. Only a few survived, including Archmaester Gyldayn and Dany's mom, Rhaella Targaryen. Rhaella apparently gave birth to Rhaegar during the fire (how??), and the prince grew up all weird and sad about the circumstances of his birth.
Some fans speculate that the tragedy was part of a Maester plot to stop the Targaryens from bringing back dragons, or that it was some other coup by the great houses of Westeros to end the Targaryen dynasty. Or maybe it was just bad fire safety and dark magic. Whatever it was, the fire at Summerhall killed off a number of Targaryens, leaving only Dany's grandparents, her parents, and her brother (and like, one random aunt who married into the Baratheon family).
By the time lil' Dany was born, Robert had killed her brother Rhaegar and Jaime had killed her dad. Her mom died in childbirth. Only Dany and her brother Viserys were left to carry on the Targaryen line. And then Viserys went and got himself all smothered to death in molten gold.
So it's just Dany out there (and maybe Jon and fake Aegon). She's the only verified Targaryen left. And if her house has any chance at all, she'd better be a whole lot smarter at playing with fire than the previous Targaryens were.