Here is the breakdown as I know it...
Double Clutching: Slowest of the shifting, was used long ago in big rigs more often than not. You start off in gear, when the time to shift comes, clutch in, pull the shifter into neutral, clutch out, clutch in, pull the shifter into gear. I guess this is used to help line the gear faces up in trannys without synchroes. Not something any new car needs, and not something used in racing.
Speed Shifting: Might even be as slow as double clutching. This is used when you have clutch issues more times than not. When the time comes to shift, you get to an even pedal pressure, not accelerating and not completely letting off. You slide the shifter out of gear without touching the clutch. Then put pressure on the next gear, and use the pedal pressure to bring the RPM easily yo the right spot (you just feel for it) and the shifter will slide into gear. Not used for racing, I have only used it for fun and when my clutch cable goes.
Regular Shifting: Used by most in normal driving and racing. With the car in gear, you push the clutch in, put the shifter smoothly into the next gear, and release the clutch. You do so by letting off the gas in between the shift, and rolling into the throttle for the next gear.
Powershifting: Used in racing only more or less. With the car in gear, get ready for the next gear with a little pressure on the shifter. When the right RPM hits, leave the gas to the floor and stab the clutch. Move the shifter (hence the pressure towards the next gear) as fast as possible, and have that clutch out before the RPMs scream to the limiter. When done right, the RPMs shouldn't climb much at all, the car shouldn't nose dive on the shift, and the timeslips should show it.
Thats about how I see it. I also prefer the last one, without it, I would probably have a mid 13 second car,

.