Cut-off mode is the BJT region where both junctions are reverse-biased. With neither junction forward-biased, there is essentially no carrier injection anywhere, so essentially no current flows through any terminal. The transistor is off.
For an npn, cut-off means the base–emitter voltage is below the turn-on threshold — practically, less than roughly 0.5 V, so the EBJ is not forward-biased. No EBJ forward bias means the emitter injects no electrons into the base, so , , . Only the tiny reverse leakage currents of the two junctions flow, which are negligible for hand analysis.
Cut-off is the open-switch state in digital logic, the complement of saturation (the closed-switch state). A BJT used as a logic switch swings between cut-off (output high, transistor an open circuit) and saturation (output low, transistor a near-short with only across it); it never lingers in active mode, because active mode dissipates power and is reserved for amplifiers. Where this sits relative to the other regions is mapped in BJT operating modes.