With single-trigger acceleration, shares become fully vested after a single event, typically an acquisition. Single-trigger acceleration is somewhat rare.
More common is double-trigger acceleration, which is triggered by two events -- usually a change in control followed by the termination of the employee.
Pros and cons of accelerated vesting
From an employee's perspective, there's really no downside to accelerated vesting beyond the potential tax consequences. You get the equity compensation you were promised on a faster schedule if a triggering event occurs. That can provide more financial security since many key employees at early-stage start-ups work primarily for equity.
It's a bit riskier from the employer's perspective, though. The reason for having a vesting schedule is to encourage employee loyalty and retention. Key employees have less motivation to remain at a company once their shares are fully vested.