On Night 1, you target the first player in the playerlist. On Night 2, you target the second player, and so on.
The effective behaviour of this power role is to add a additional Night phase immediately after each regular Night phase, on which only the targeted player can act. However, instead of actually running through both Night phases, the targeted player is informed that there might be an additional Night phase in which only they can act, and asked to send in their actions at the same time as their actions for the regular Night phase. If the action succeeds, the additional Night phase will be simulated as Instant Night.