In season 5 episode 9 of Game of Thornes, The Dance of Dragons
Arya was instructed to kill The Thin Man with poisoned oysters.
But in episode 10, Mother's Mercy
Arya uses a face from the face vault and kills Meryn Trant instead. She is later punished for taking a life that "was not her's to take."
Who decides which life's can be taken and who is allowed to take them? Who decided The Thin Man was allowed to be killed?
TL;DR : The Faceless Men are a guild of HIRED assassins and usually kill by contract.
While it is true that The Faceless Men are a religious group that worships The Many-Faced God, they are also considered the world's foremost assassin's guild. What they worship is death, and assassination is their prayer.
There are several references in both the book and show to this. When discussing the potential assassination of Daenerys Targaryen, the Small Council considered hiring a Faceless Man. Petyr Baelish, the Master of Coin, balked at this idea and stated the following:
PETYR: Do you have any idea how costly they are? You could hire an army of common sellswords for half the price, and that’s for a merchant. I don’t dare think what they might ask for a princess.
In addition, GRRM himself was asked during a chat discussion about the price of hiring The Faceless Men. He responded:
The Faceless Men don't post a list of prices on their door. The way it works, you go to them and tell them who you want killed, and then they negotiate the price. The more prominent the victim, the more difficult to get to, the more dangerous for the assassin and the guild, the higher the price.
Note that the price is not always money - it can also be something you own, or even your child.
In addition to contract assassinations, their temple - The House of Black and White - in Braavos has numerous fountains which are tainted with an odorless & tasteless poison. Any person who wishes a painless death may enter and drink from the fountains, with the only price being that their face becomes the property of The Faceless Men. This opportunity is called "The Gift", and it is how the guild obtains the faces used by its members.
