There are a ton of people vying for the Iron Throne on Game of Thrones, so what, exactly, makes Daenerys Targaryen's claim so strong? Other than her badass dragons, her fierce coat collection, and the fact that she might be the long-awaited Prince Who Was Promised? Her blood, of course.
Like Westeros's current queen, Cersei Lannister, Daenerys comes from a long line of noble blood. She's the only surviving child of the last Targaryen ruler of Westeros, King Aerys II. As we've been reminded over and over again in season seven, Dany's late father is also known as the Mad King, since his increasingly erratic reign ended with the gut-wrenching and unnecessarily cruel deaths of Rickard and Brandon Stark (aka Ned's father and brother). After their executions, houses Arryn, Baratheon, Stark, and Tully rebelled against Aerys and took over King's Landing.
Before the Targaryen king completely lost his mind, he wed Rhaella Targaryen, his sister, because the Targaryens are really into incest (they wanted to keep their bloodlines completely pure). Rhaella was deeply unhappy about being forced to marry her brother, but together they bore three children: Rhaegar, Viserys, and Daenerys.
By the time Dany was born, her father was already too far gone. The violent rebellion reached King's Landing, and while Rhaegar, Dany's eldest brother (and Jon Snow's father), fought back against Robert Baratheon (and died in the process), Aerys sent Viserys and a pregnant Rhaella to Dragonstone for safekeeping. Rhaella gave birth to Dany on the island in the middle of a huge storm but passed away due to complications during the birth. The Mad King also died in the Sack of King's Landing, murdered by the captain of his own Kingsguard, Jaime Lannister.
Fortunately for Viserys and his newborn baby sister, the pair were smuggled out of Westeros to the Free Cities across the Narrow Sea by Targaryen supporters, where they remained throughout childhood. Now that Dany is vying for the throne her father once occupied, it's up to her to break the chains and create a new legacy for her family rather than becoming a mad ruler herself.