Granny shifting is shifting at very low rpm, driving slowly, like a stereotypical granny would.
Double clutching is a technique used for rev-matching in downshifts. It is useful for older cars, with not very good synchros. What you do is press the clutch, disengage the gear you are in, release the clutch while in neutral, step on the gas briefly to get the engine rpm up, press the clutch again, engage the lower gear, release the clutch, and then accelerate. In more modern cars you can do the throttle dip without releasing the clutch, since the cars have better synchros. It is essentially like heel-and-toe, but without braking. The benefit over not rev-matching while downshifting is that you can release the clutch more rapidly and get on the power quicker, because the engine is already closer to the rpm it should be before you release the clutch, instead of having the clutch speed up the engine while slowly releasing the clutch. It is useful when you want to make a fast overtake on the road and want to get on the gas as quickly as possible. If you get the rpm matched perfectly, you can do the downshift without even using the clutch, but DO NOT TRY IT.
You can look up videos on Youtube that show it being done, as it is a lot clearer what the process is than reading my explanation. This is how I first learned about it.
Hope this helps.
I drove without touching the clutch quite often. Its a great joy to be able to accomplish it :)
There should be medals for this.
Granny shifting is short shifting. And double clutching has been phased out. On very old heavy duty trucks you had to clutch, shift to neutral, release the clutch, wait for the synchros to sync up and then do it again. Wanna be boy racers seem to think you still need to do it in passenger vehicles.