In the English language this term unfortunately describes the process of rev matching in both upshifting and downshifting.  I'd like to distinguish between them.  Let's use two different terms:

You do "double clutching" on the upshift:

lift off throttle -> depress the clutch -> switch into neutral -> release the clutch -> depress the clutch -> switch into a higher gear -> release the clutch -> step on the gas.

You do "double declutching" on the downshift:

lift off throttle -> depress the clutch -> switch into neutral -> release the clutch -> blip the throttle, increasing rpm in order to match the higher rpm of the secondary shaft in the gearbox needed for the bigger diameter lower gear -> depress the clutch -> switch into a lower gear -> release the clutch -> step on the gas.

Blipping the throttle to precisely match the rpm is no doubt an acquired skill, but if done properly it affords a silky smooth downshift.
 

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