Prompt skill is a type of mystery that shows in its description. These skills will be used when you press the *on. Skills that do not have prompt will be cast in place of the character's normal attack that turn.
For example, if I use a mystery at the start of turn 2 it will be used before anyone starts attacking normally. If I use a mystery that does not have prompt, then I have to wait until its that character's turn before they will use their mystery.