The black and white syringes refers to a set of two syringes. While one is white, the other is black. It is essential to inject a person with them before they can be put in a simulation. The black injection makes the subject forget their memories, and the white injection brings them back.
Henry creates an entire loop in the simulation to get the Key from Maura. Every time the loop restarts, he needs Maura to forget the previous one and feel like she’s on the ship for the first time. To accomplish this, Henry gives his daughter the black injection, which makes her forget the previous loop and wake up on the ship. Henry uses the white injection too, but just on one occasion - when he wants to show Elliot the truth and make him remember what his mother made him forget.
In Elliot's memory he sees Maura walking toward him with a black injection, and before she gives it to Elliot, she says that he’ll forget what happened to him in the ward. Maura used the black injection to make him forget that he was ill and dying.
Almost all the passengers of Kerberos don’t remember how they came onto the ship, and it is likely that Henry injected them with the black liquid too.