Vesting acceleration provisions enable vesting to be fast-forwarded when certain conditions are met. There are two broad categories of vesting acceleration: double-trigger and single-trigger. The number of triggers refers to the number of events that have to occur in order for the acceleration to kick in.
If someone has double-trigger vesting acceleration (commonly referred to as double-trigger acceleration) on their stock or stock options, that typically means the vesting will accelerate if (1) the company is acquired and (2) that person is terminated in connection with or following the acquisition1. This is the most common type of double-trigger acceleration, although there are many other possibilities. For example, some double-trigger acceleration provisions only allow for the acceleration of vesting for a certain percentage of the shares. Or, more rarely, the triggers themselves may be different.
The most common form of single-trigger vesting acceleration (commonly referred to as single-trigger acceleration) is for vesting to accelerate if the company is acquired. A far less common form of single-trigger acceleration is for the vesting to accelerate if the stockholder is terminated.
Double-trigger acceleration is standard for stock issued to founders, and is occasionally used for executive-level hires. It is not typically given to other employees or consultants, because it is viewed as undesirable by acquirers and consequently VCs. Naturally, single-trigger acceleration is viewed as even more undesirable. Single-trigger acceleration is usually only given to advisors, if at all.
You should consult an attorney before giving anyone any form of vesting acceleration. Aside from double-trigger acceleration for founders, vesting acceleration is uncommon and should be used with caution.