Double clutching is generally used on downchanges, but it can be used on upchanges on very old cars where the gearbox synchros are pretty knackered. The basic principle is to match the speed of the gearbox input shaft to the speed of the output shaft, and reduce the load on the synchros.
Most common on downshifts, there are 2 variants - heel/toe and double clutch.
In a heel/toe, most common on modern cars to prevent the rear wheels trying to drive the engine when you engage a lower gear, the idea is to blip the throttle in the middle of the gear change, while the clutch is depressed and the gear lever is in the neutral position between gears. This brings the engine speed up to the match road speed of the lower gear. It is usually done by applying brake with the toe of your right foot, and pivoting the heel of your foot around to blip the throttle, hence its name - heel / toe.
With double clutch, instead of having the clutch depressed, you release the clutch while blipping the throttle, then depress it again to engage the gear. This is to match the engine speed to the road speed when the synchros are crook and don't actually do the job they are supposed to.
Hope that all makes sense.