I'm going to expand on rpmerf's answer, where he has disambiguated between Heel and toe and clutchless shifting.
I'l briefly explain the driving technique only to better illustrate what it hopes to mechanically achieve. And so, please excuse me if get the nuance of the technique wrong.
Here's an image of a gearbox. I couldn't find a one with synchromesh, just assume that it goes where the DOG GEARS are.
HEEL AND TOE - This is a driving technique for DOWNSHIFTING while BRAKING.
The point of doing this is REV-MATCHING the engine with the new transmission speed after engaging a lower gear and doesn't help the synchromesh unless you're DOUBLE CLUTCHING as well.
- You press down on the brake (right foot), modulating it as per requirement
- You press down on the clutch (left foot) and shift to a lower gear
- You pivot your braking foot (toe still on the brake), such that you have your heel to press down on the accelerator (clutch is still disengaged/disconnected). You press it enough to roughly bring it to the rev's required for the lower gear.
- You let go of the clutch (modulate)
- You still continue to brake as per requirement
point 2 . While downshifting, let's assume that the time spent on moving the gear lever is negligible, the engine needs to pick up speed to match the rotational speed of the gear shaft, which is , for the same road speed, now higher than the previous gear. Slowly engaging the clutch would have done the same thing. By applying a commensurate throttle, you've aided this rev-matching process which was left to the clutch alone, and completed the shift in an admitidly shorter time. -
Faster shift - lesser time spent in between gears is good for all sorts of reasons, one of which is having consistent engine braking.
Alleviate clutch abuse - Apart from wear, you significantly reduce driveline shock that can result from quickly releasing the clutch.
Everything above apply's when you just want to shift down and don't have a need to brake. All you do is omit the braking part, and keep the rev matching part i.e clutch , rev-match and shift, release clutch.
This is what you want to do if -
1. you want to reduce your synchro wear
2. your synchro's are already worn
3. your vehicle doesn't have synchro's (old vehicles, and some between the 1st and 2nd gear) . No synchros on reverse gear either. but since one is stationary while engaging reverse, you're already rev-matched. Try reving in neutral and quickly engaging reverse and you'l hear a grinding noise.
By double clutching, you're trying to REV MATCH the lay shaft/neutral shaft, which on a constant mesh gear box is constantly meshed with all your gears, to the OUTPUT SHAFT , which is connected to the synchromesh and mates to the selected gear.
Drop to neutral.
- Release the clutch while modulating the accelerator (reducing it, though not all the way) such that the new (lower rev) of the gearbox neutral shaft is the rev matched to the higher gear, connected to the driveshaft.
Press clutch and shift to the new gear.
Admittedly, this process can be made faster (driving technique nuances) , taking into account that simply by lifting off the throttle you're allowing the engine rpm to fall, and hence, double clutching while lifting off the throttle and getting a hang of timing can expedite the process.
If performed while braking, it'l be DOUBLE CLUTCHING WITH HEEL AND TOE. i'm only going illustrate the double clutching process.
- Press clutch
- Drop to neutral
- Modulate throttle - Increase rev to match the lower gear you want to shift down to . You're bringing up the neutral shaft to the RPM expected by the new gear hence doing the job of the synchro.
- Press clutch - so as to disconnect the engine again and have the impending shift occur without the additional engine inertia to deal with , as rev- matching can never be truly perfect.
- Shift gear
- Release clutch - Re- Engage the engine to the transmission.
Again , in practice, it's easier to blip the throttle than to hold a constant rpm when the engine is free- revving. Hence, depending on the speed of the shift, one may
- Blip once - at point 3. and finish the rest of the steps quicky before the engine drops in RPM .
- Blip twice - at point 3 and point 5 if the shift is slow . - long gear levers in trucks, or just a lazy shift.
A clutchless shift isn't commonly done on a transmission with synchro's. It can lead to premature synchro wear even if the shift feels smooth.
Clutchless shifting can be done in CONSTANT MESH GEARBOX with dog gears both up and down. It helps if the shifting mechanism is SEQUENTIAL and the rotational inertia of the transmission is low. Though, all that's another topic.