First, a simplistic diagram of your car's drivetrain to make all this a little simpler:
---------- driveshaft ---------- layshaft ----------- to wheels
| engine |------------>| clutch |---------->| gearbox |<---------->
---------- ---------- -----------
For gearboxes created before the invention of synchromesh gearboxes, double clutching was neccessary in order to speed up the layshaft to the speed of the driveshaft (essentially, to get the gears up to the same rpm as the wheels). If you didn't do this (or if your syncros are going out in your modern gearbox), you'll hear an unpleasant little grind as you jam the gearbox into gear. That grind is the layshaft (the shaft which does the actual shifting mojo), which is probably rotating pretty slowly by the time you reengage the clutch, meeting the shaft going to the wheels, which is probably rotating at a pretty good clip -- equal to the rpm of your wheels. The two gears grind a bit as one speeds up to the other's rpms. This is a bad thing, since it both makes you look like a jackass to passengers and pedestrians and wears out whatever gear you just jammed into. These are the steps to double clutching:
- Disengage the clutch (Push the pedal down): This disengages the gear box from the driveshaft (the engine) and allows you to...
- Pull the gear into neutral
- Engage the clutch briefly: This doesn't transfer any power to the wheels, but merely gets the gears up to wheel speed. You might also have to nudge the gas a bit to get the speed right, depending on...a whole lot of stuff, actually. It really depends on your specific car and the situation (speeding up / slowing down, braking, shifting up / down, etc.)
- Push the gear into whatever gear you want: Since the gears are now moving at approximately the same rpms as the wheels, there's only going to be a little grind as the two spinning gears meet and mesh.
- Re-engage the clutch and go, baby, go!