Night Specific (or Night X) is a role modifier that enables a power role to act on specified Nights only. A role could be limited to one night only or a range of nights on which they can act. Typically this modifier is attached to roles as a method of weakening them, reducing the amount of times in a game they can use an action, similar to Even/Odd Night.
The Day version of this role is Day Specific, limiting the use of a power role that has effects during the Day phase onto specific Days.
An example of use is the Diffusion of Power setup.