Flight Level Change or Vertical Speed - When to use it?
FLC mode maintains airspeed during a climb or descent, while VS mode maintains a specific vertical speed.

Often air traffic controllers will request that you "maintain 250 knots in the descent" or something to that effect, which is much easier to achieve when using Flight Level Change. As another answer points out, by maintaining airspeed, climbs are made safer as the autopilot will not attempt to maintain a rate of climb regardless of whether or not the airspeed is safe.

FLC mode is typically only available in advanced autopilots, and as far as I know they need to connect to an Air Data Computer to get the necessary information.

How to use FLC would depend on the equipment available to you: 


aircraft with auto throttle will maintain speed and an appropriate rate of descent automatically. 
for aircraft without auto throttle you'd set your target altitude, enable FLC, adjust your target speed if needed, and set your throttle for climb or descent to maintain the desired altitude change rate.


I'm referencing the Garmin GFC700 as it's the only autopilot I've used with FLC mode, but I imagine the same steps and rules would apply.