Invade the Surface: After an action resolves in which Aquaman moves due to his own action, you may give any number of other friendly characters possessing or a free action to move up to three squares if they currently occupy water terrain.

Flood the Land in Revenge: When Aquaman is KO'd, all grounded terrain becomes squares of water terrain for the rest of the game instead of squares of clear, hindering, or blocking terrain, even though this character is not on the map.

Underground Geyser: Give Aquaman a double power action and choose any square of clear terrain within a range of 8 and line of fire. Place a special terrain marker in the chosen square and it becomes water terrain for the rest of the game. Deal a character occupying that square 3 damage, and then each character occupying an adjacent square is knocked back from the chosen square 3 squares.

All Must Be Submerged: Aquaman can use Toughness and Willpower.

When Flood the Land in Revenge is triggered, what happens to objects that are on grounded terrain?
Normally, an object occupying a square would make that square hindering terrain. And if there was another terrain type also in that square (say, you TK an object out into water terrain), then that square will share properties of both hindering terrain and the other terrain type.

However, in the case of this trait, the square becomes water terrain instead of the other terrain types listed. The object will stay in that square, nothing says you remove it or destroy it, but the square itself will be considered water terrain for movement and line of fire purposes. The only effect the object will now has is it may still be picked up and held with Super Strength, targeted by TK, or used with game effects that require an object (and not hindering terrain).

What is this double power action thing that Underground Geyser is talking about?
See the Double Power Actions entry for more information.

When is the damage dealt by Underground Geyser? Is a character always damaged any time it occupies that square? Or only when this power is first used? Similarly, when is knock back determined?
Damage and knock back are determined when this power is first used. After the power is used, the chosen square is simply a square of water terrain with no additional effects.