Cascading elevators can be driven by cables/cords, chains, or belts (we'll call them all "cables" for brevity). They use multiple lifting "cables" to move all of the stages.
The first stage is driven by the motor(s) with any method that is desired (winch, chain sprocket, rack and pinion, lead screw, etc.).
The higher stages are each driven off of the previous stage in a way that causes each stage to move (relative to the previous) the same amount at all times
Cascading elevators can also be equipped with retracting "cables" on each stage (tied to the previous stage) to provide powered motion in the downwards direction.
Cascading elevators are more complex than continuous elevators, but are more forgiving in the implementation.
They generally have less stroke length than a continuous elevators and cause the COG of the robot to be higher throughout the motion of the elevator.
Continuously-rigged elevators use a single lifting cable to move all of the stages. The uppermost stage moves first and none of the lower stages move until that stage has reached its end stop.
Continuous elevators can also be equipped with a retracting cable to provide powered motion in the downwards direction. This cable is just connected to the last stage and routed straight down.
Continuous elevators are mechanically simpler than cascading elevators and can provide addition stroke length, however they are typically difficult to implement in practice due to various sources of binding that are inherant to the design.