There are two modes in
Aquattack, Arcade Mode and Challenge Mode. Challenge Mode consists of 50 levels where you have to reach a certain goal, such as earning a set number of points or getting certain combos within a set time limit. This is a great place to figure out what all of the various obstacles and power-ups are in the game, since they explain what your goal is before the level begins, but they don't really do that in Arcade Mode and you are left to figure things out on your own.
In Arcade Mode, your goal is to clear fish from the board by matching 3 or more in a row going either up/down or left/right, but not diagonally. The further you progress in the game, the more colors and patterns of fish will emerge, making it more difficult to match them up. Also, obstacles like coral will appear in certain spots and won't be able to be moved the way the fish can, by simply moving them up and down and side to side. In fact, the coral will even render entire rows unable to be moved and you'll have to utilize mines to blow up the area around the coral in order to free up the row.
There are also helpful power-ups like an electric eel and the above mentioned mines that will clear large areas of fish, not by killing them, but by making them swim away in a very family-friendly way. You can also get special items like rainbows or starfish by accomplishing tasks like matching a row of 5 fish.
You start the level with one minute on the clock and the timer is constantly ticking down, however for every fish you clear, you get seconds added to the timer. So even starting with one minute, you can actually manage to play for a very long time.