Double action (also two-stage) is a provisional term for a switch with two sets of contacts with independent actuation staggered by force. The first set of contacts actuates first, and the second set actuates when the switch has been pressed more firmly.
One suggestion is that such switches assist with the activation of an auto-repeat (typematic) mechanism without needing to rely on a timer. Instead, key repeat is based on switch pressure. In the case of the Alps SKCLKB, the first set of contacts actuate at 88 cN, while the second set actuate at 490 cN, allowing the key to function normally when struck (without unintended repeats) but enable the repeat circuitry when held firmly.
Variations of double action press mechanisms are also found in electronic claviers, where the time between actuations is used to determine the speed — and strength — of the note. However because clavier keys are levers, the mechanism is often instead implemented as two simple momentary switches underneath the lever.