After two long seasons of training with the Faceless Men, Arya finally showed her badassery in Game of Thrones' season 6 finale, when she snuck up on Walder Frey (but not before feeding him his two sons inside a nice warm pie).
At the beginning of Season 7, she poisoned the entire male half of the Frey dynasty in one sitting. Now that she has gone full assassin, Arya can begin her quest to complete "The List," her collection of people who she wants to see dead one way or another.
The list itself appears to change with time, and she's killed at least two of them herself so far, while The Hound may have earned himself a reprieve:
So why were these names on the list to begin with?
1. Cersei Lannister, for her role in the deaths of her father Ned Stark and others (ALIVE)
2. Joffrey Baratheon, for ordering the deaths of her father and Mycah the butcher's boy (KILLED—poisoned by Olenna Tyrell)
3. Ser Ilyn Payne, for wielding the sword that killed her father (ALIVE but kind of forgotten about)
4. The Mountain, for leading the brutal tortures at Harrenhal (ALIVE, sort of)
5. The Hound, for killing Mycah on Joffrey's orders (ALIVE, though she left him for dead, and her Faceless mentor Jaqen accused her of lying when she told him she wanted him dead)
6. Melisandre, for taking Gendry away (ALIVE)
7. Beric Dondarrion, for taking Gendry away (ALIVE, sort of)
8. Thoros of Myr, for taking Gendry away (ALIVE)
9. Tywin Lannister, for leading the Lannisters against her brother Robb (KILLED—shot by Tyrion Lannister on the bog)
10. Ser Meryn Trant, for killing her master Syrio Forel and helping Cersei's coup (KILLED—eyes stabbed out then throat cut by Arya)
11. Walder Frey, for his role in the deaths of her mother and brother (KILLED—throat cut by Arya)
Arya's list contains those whom she knows have committed crimes against her family. Littlefinger might have technically escaped the list, but he recently met his end at her hand, all the same.
It could be said that Melisandre, Bericm and Thoros have been phased out of the list, especially as one of the main reasons she added them was that they released The Hound, whom she has now seemingly forgiven.
Meanwhile, Polliver and Rorge were also briefly members of the list, for killing her friend Lommy and threatening to rape her, respectively. Arya got her revenge on both of them eventually, killing them herself while on her travels with The Hound.
She also asked Jaqen H'ghar to kill torturers The Tickler and Ser Amory Lorch, after he required her to give him three names to appease the Many-Faced God once she freed him. Her third name was due to be Tywin Lannister, but she changed it to Jaqen himself in order to make him agree to help her on her next journey.
The late Roose Bolton was never on the list, as she was unaware that he was present and heavily involved in the Red Wedding. The same goes for dead Ramsay Bolton's killing of Rickon and abuse of Sansa. She is also seemingly unaware of Theon Greyjoy's taking of Winterfell and supposed killing of Bran and Rickon.