In Brawl and Gauntlet, Action Points are the orange blocks displayed in the gauge on the bottom right of the screen. The gauge becomes full at 5 points, which is the maximum you can have at a time.
Action Points allow you to use your dragons' special abilities (2nd and 3rd ability slots). Different abilities will have different Action Point costs, displayed in the bottom left of the ability icon. Note that basic attacks (1st ability) do not cost Action Points.

You earn one Action Point on each one of your turns. You can allow them to accumulate to use more powerful abilities or an ability from a specific dragon. In Brawl specifically, if the match extends past a certain amount of turns, a x2 Action Points modifier will be applied, and both sides will earn double the action points from each of their turns.
Additionally, when a dragon is eliminated, an Action Point is gained by the side who lost the dragon.