Arya originally puts Sandor Clegane, nicknamed "The Hound," on her list for killing her friend Mycah, the butcher's boy, in season one. It happens during Joffrey and Cersei's trip to Winterfell, where the former provokes an argument with Arya and Mycah that ends with Arya's pet dire wolf Nymeria biting Joffrey when he tries to attack them. Joffrey lies about what actually happened to his mother so she orders Sansa's dire wolf Lady executed, and Sandor Clegane chases down and kills Mycah (he remarks to Ned Stark that "he ran, not very fast").
Over time the Hound comes to prove himself as much more sympathetic a character than he initially seems, and comes to be an ally for both Sansa and Arya at points. In season three he ends up capturing Arya after she flees from the Brotherhood Without Banners when they sell Gendry to Melisandre. Their relationship is prickly, but Arya begins to come around to the gruff warrior after he tells her he saved Sansa from a group of would-be rapists, and also plans to deliver Arya to her mother and brother rather than to Cersei in King's Landing. They come upon the Red Wedding massacre and he helps her escape unharmed.
In the end, though, Arya fulfils her promise to kill the Hound . . . well, sort of. When he gets into a brutal tussle with Brienne, he falls from a cliff and seems not long for the world. Arya comes upon him and he tries to goad her into killing him and putting him out of his misery, but instead she takes his coins and leaves him to slowly die from his wounds. However, in season six we discover that Clegane is saved by a group of peaceful villagers who take him in. Arya also later mentions that she's eliminated him from her list anyway, since she felt conflicted about letting him die.