Action Count is the number of commands the player can perform when their active turn rolls around.
The Action Count recovers to it maximum value during each new turn. If you end the turn with Action Count points remaining, those Actions points will not be carried over. The Action Count's maximum value varies with the equipped Mic's proficiency.
By defeating MAD, achieving Overkill, etc., you increase the Action Count for the next turn. The Action Count can increase up to as much as twice the current maximum. The Action Count resets when the battle ends, and will not carry over to the next battle.