Warning: Spoilers for Game of Thrones ahead!
As the final episodes of Game of Thrones continue to unravel, more and more characters who haven't met before are finally coming face to face. In the season's second episode, another new pair of characters finally met: Jaime Lannister and Daenerys Targaryen. The Lannister knight, known as the Kingslayer, comes to Winterfell to keep his promise to stop the White Walkers, regardless of whether or not Cersei follows through with her part of the deal. Dany, however, is not pleased to see him, and is reluctant to do anything except condemn him as a criminal — and she's got a pretty good reason.
Before the series began, as viewers will remember, the Targaryens ruled Westeros, and Jaime was in the Kingsguard of Aerys II Targaryen, Dany's father. However, Aerys had a more popular moniker, the Mad King. Aerys reportedly suffered from a gradual onset of madness, in large part due to the Targaryen tradition of close-relative incest. During the rebellion led by Robert Baratheon, Aerys was killed when Jaime turned traitorous and personally murdered him, paving the way for Robert to claim the throne (and for Jaime's sister-lover, Cersei, to marry the new king and be queen). This earned him the nickname of "Kingslayer," which has followed him ever since.
Although he did turn against the king, Jaime's killing of Aerys wasn't entirely unjustified. During the war, Aerys realised he had been betrayed by the Lannisters and turned to a plan he had hatched long ago. After developing an obsession with wildfire during his decline, he had placed the magical weapon all over King's Landing. When he learned of the Lannisters' betrayal, Aerys intended to set off the wildfire, burning the city and everyone in it, believing he would come through the flames as a dragon. Jaime murdered Aerys to stop this murderous plot — but, in a cruel irony, Jaime's own sister eventually carries out the same plan. In the sixth season, Cersei sets off some of the wildfire to destroy the Great Sept and everyone in it, including religious leaders and the entire Tyrell family, gathered there for Cersei's trial and that of Loras Tyrell.
Dany is ultimately convinced to allow Jaime to fight with her army thanks in large part to Brienne's testimony, although it's clear she hasn't forgiven him for killing her father. Even so, Jaime is a different man now than he was long ago. Whether or not this comes back into play as the final episodes pan out, we can only wait and watch.