Double-trigger acceleration is a term used in the context of vesting equity (shares or options) in a company. Vesting of equity can be accelerated on agreed-upon terms.
Double-trigger acceleration of vesting is based on two events — for example 1) an acquisition or other change of control and 2) termination within a specific time period (generally without cause). If both events occur, the individual's equity is accelerated to completion.
An alternate type of acceleration is single-trigger acceleration, which accelerates vesting on only one event (typically acquisition / change of control).
Double-trigger acceleration is more common than single-trigger acceleration. It helps ensure employees don't leave upon an acquisition or change of control since they would not be fully vested. But it also protects those employees' interests by allowing them to vest completely if the acquiring company chooses not to continue to employ them after the acquisition.