This battle mechanic will usually having the player hit an additional button after selecting a regular attack or special move to cause the attack to hit, deal more damage, or any other number of effects. There almost always is an indicator to tell the player when to hit the button, but that doesn't always have to be the case. Some games that use a reaction system have don't only use it for offensive or even inputted actions but also when the player is defending as well. This usually appears in games that apply the system to all ranges of it's battle system.
This is somewhat similar to the Quick Time Event mechanic, but used within a turn-based battle system.