Amp Damage is damage that the shields add to your first shot when your shield has full capacity. So you can do a lot more damage with amp.
The first shot you take with a fully charged shield gets the bonus Amp Damage added to it.
When the shield recharges, it will add the amp damage again, and so on, every time the shield has charged to a full capacity
The Amp Damage used Amp Shot Drain which is subtracted from your shield's charge.
An example shield with Amp Damage: The Bee