What Happens When You Betray the Faceless Men?
As per usual, this week's Game of Thrones left us wondering what lay in store for many characters, but no one's fate seems more up in the air than that of Arya Stark. She fails to assassinate the actress Lady Crane and appears to have fled the House of Black and White. She also reclaims Needle, the one real possession she still had from her "old" life. It seems that she's given up on becoming one of the Faceless Men, but they won't be letting her off so easy.
The Rules of the Faceless Men
The Faceless Men serve The God of Death, also called the Many-Faced God. These assassins have a serious respect for death and killing and see themselves as servants of death itself. When Arya meets Jaqen H'ghar on the way to the Wall in season two, he says he will kill three people for her to make up for saving his life and the lives of his companions, which essentially "kept" those deaths from the Many-Faced God. And as he says in season five, she doesn't get to decide who dies just because she acts as a Faceless Man. The Faceless Men consider their task of ending lives as gifts asked for by (and which must be given to) the Many-Faced God as a way to end suffering.
Anyone who joins the Faceless Men must give up their old life and identity completely in order to become "no one." That way they can use the faces collected in the Hall of Faces to do their work for the Many-Faced God. Until someone completely gives up their old life, they can't truly become "no one" and a true Faceless Man.
Betraying the Faceless Men: No "Third" Chances
While becoming "no one" and being accepted as a Faceless Man is more of a prerequisite requirement (it takes a while for Jaqen H'ghar to be convinced that Arya has indeed completely let go of her old self, and the Waif is never fully convinced) than a rule, granting death is a whole different story. And while Jaqen doesn't specify the exact punishment for breaking the rule about taking lives for personal reasons instead of in service to the Many-Faced God, we soon see that it's pretty severe.
When Arya kills Meryn Trant out of her own hatred for him at the end of season five instead of the man she was sent to assassinate, she's punished by having her sight taken and being cast out of the House of Black and White. (In the book A Feast For Crows, Arya does disobey the Faceless Men by killing someone and loses her sight, but she's still allowed to continue her training at the House of Black and White.)
There also seems to be a little wiggle room on the permanency of punishments in the TV show — at least for Arya. Jaqen H'ghar clearly thinks Arya is worth giving a second chance, and so in season six he restores her sight and lets her begin training again. She commits with seemingly renewed devotion, and finally Jaqen gives her another chance to give the gift of death and prove herself by killing Lady Crane. He ominously adds that she won't be granted a third chance and that a new face will be added to the Hall of Faces "one way or another."
What Does It Mean For Arya Now?
When Arya again fails to complete her task of granting death in "Blood of My Blood," she doesn't even try to lie about it or make excuses. In fact, it looks like she doesn't even go back to the House of Black and White, because she knows what fate awaits her. Jaqen H'ghar sends the Waif (who learns of Arya's failure by spying in the theater, because of course she does) to find Arya wherever she's hiding and kill her. Like Jaqen said, there has to be a face added to the Hall. The Many-Faced God asked for a death, after all.
However, the Arya Stark we know is back to embracing her true self from the looks of it, and the Arya Stark we know is a total BAMF. Furthermore, not only is she now more skilled in martial arts after training with the Waif, but she also has her trusty Needle back. So while her death seems to be the next intended gift to the Many-Faced God, it's clear Arya has no intention of going quietly.