The Duel in Braavos is a battle between Arya Stark and the Waif after Arya fails to kill Lady Crane. The Waif has a conversation with Jaqen H'ghar, suggesting that they had a deal allowing the Waif to kill Arya for her failure. After hearing Jaqen's request to kill Arya humanely, the waif sets out to fulfill her mission.
The Waif eventually manages to locate Arya, and approaches her wearing the face of a feeble old woman. Before Arya can react, the Waif pulls out a dagger and repeatedly stabs Arya in the stomach, critically wounding her. Arya manages to momentarily escape by jumping into the canal, though her wounds begin to bleed out. Arya stumbles to the acting tent where she is found by Lady Crane, who then takes Arya and heals her at her home. 
The Waif comes the next day and kills Lady Crane, who had been sheltering Arya. She proclaims that Lady Crane's death would have been painless had Arya succeeded in her mission, and begins to pursue an injured and fleeing Arya throughout Braavos. Arya leads the Waif to her hideout, where she finds Needle and cuts the candle that provides the only light in the room. Arya, having experience in fighting without sight, is victorious, and adds the Waif's face to the Hall of Faces.
Aware that Jaqen sent the Waif to kill her, Arya points Needle at him in an accusatory manner, though Jaqen claims that she has finally truly become "no one", indicating the completion of her training as a Faceless Man. In reply, Arya reaffirms that she will never be "no one": She is Arya Stark of Winterfell. Lowering her sword, she then leaves the House of Black and White and Jaqen for good, making the ultimate decision to return to Westeros, her home. Despite forsaking the Faceless Men, Arya does not forsake her training; she uses her new abilities to disguise herself as a servant of Walder Frey in order to kill him and two of his sons.