Counter Attacks are one of the most useful features in the game. A Counter Attack happens during your opponent's Power Shot. If you are able to Kick the ball right as it's about to hit you, your Character will counter it. If you counter someone's Power Shot, then you will use their Power Shot unless you have a character with an own unique counter attack (e.g. Austria, Uruguay and Denmark). A sparkle will appear in front of your character if you do a Counter Attack and then you will shoot the Counter Attack.
Most Power Shots aren't that difficult to counter. The ones that travel in a straight line just need a right timing. For most Power Shots you just need to: Jump up so that the ball will hit you, but right before it hits you hit the kick button and you should counter the Power Shot. If anything gets in the way of a power shot, it can't be countered anymore. Examples are rocks (caused by France's power shot), or a character, which can happen in the Cyborg's power shot.
There is an Achievement that is to win against a character with at least 5 counter attacks. The best way to do this is to use Italy, but try to use him for characters after Asura, because they have better timing and they counter Italy more often. The reason is that the defender should be able to counter Italy's Power Shot, sending it back at you to counter. If you use Italy then most of the time you don't even need to counter the defenders actual shot, because you can just pass Italy's Power Shot back and forward. Also Romania's Power Shot and Counter Attack is often countered by your opponent, so you can use him instead if you already unlocked him. Against Greece and almost all characters after him however, when the ball is countered, it will be shot in a new shot, that is an own Counter Attack of that Character. You should try to counter this in the exact same way as you counter Italy's, the ball is just moving faster.
There are certain Characters which you can unlock by gathering Counter Attacks. Here an overview of them: