Eryck,
If you are under 40, it would be rare to have learned these unless you grew up on a race track or an 18-wheeler. Others will probably add to or correct my comments, but I will give it the first shot.
Double clutching is the technique of momentarily letting the clutch out with the car in neutral between shifts, (depress clutch, shift to neutral, release clutch, depress clutch, select new gear, release clutch) and this can be done up or down in gears. It was originally done with non-synchronized gear boxes and had the effect of getting the gears and engine to approximately the same speed so that when you then select the next gear there is much less strain on the gear box. I actually don't use this, but I use a short cut that has most of the same effect, I call it "matching revs" and it involves either blipping the throttle or trying to accurately match the RPMs to where they will be when you let the clutch out in a new gear. By example, if you are in 3rd @ 3,000 RPMs entering a turn that you know wants 2nd gear, you can depress the clutch, raise the RPMs to 4,000 and let out the clutch with just about no strain on the gear box and no undue braking.
Now, if you want to get fancy, you can brake at the same time and accomplish a heel-toe shift. Entering that same turn in 3rd with your toe (really the ball of a foot as big as mine) on the brake, you blip or depress (quickly but partially) the throttle with your heel (really the outside of your right foot) to match what you anticipate the revs to require in the downshift, at the same time depress the clutch, and at the same time grab a handful of second, and at the same time let out the clutch, and at the same time you roll your foot off the brake and get back on the throttle in the new gear and bingo, you are there. Well, OK, you really can't do it all at the same time put the quicker and smoother you can accomplish this the better. Done right you avoid the more dramatic engine braking (not to mention the clutch and transmission strain) and you can negotiate the turn with little or no dive and much smoother transfer of weight from front axle to rear, making for better turn in going into a corner and controlled understeer on exit.
That a layman's explanation, there are probably some real drivers who can clean up my attempt,
Best,
g
'64 230sl, fully sorted out...ooops, spoke too soon