As everyone knows Earth is revolving around its axis from West to East. Suppose you have a train in the Northern hemisphere that is moving (for the sake of simplicity) along a meridian from South to North. Because of the revolution of the Earth, the Eastern rail (right rail) of the train will wear faster than the other rail. This force that wears the right rail of a train in the Northern hemisphere is called the Coriolis force. In the Southern hemisphere the Coriolis force is opposite, it wears the left rail of trains.
Now, suppose you have a wind blowing from a high pressure area to a low pressure area (a cyclone) from South to North in the Northern hemisphere. Because of the Coriolis effect (the force that always acts to the right in the Northern hemisphere) the air will start to rotate counterclockwise instead of following a straight line between the two areas of different pressure. If the wind is blowing from a high pressure area to low pressure area (an anticyclone) it will rotate clockwise (because of the Coriolis effect).
In the Southern hemisphere, because the Coriolis force acts always to the left, the rotation directions of the cyclones and anticyclones will oppose the directions they take in the Northern hemisphere (cyclones rotate clockwise and anticyclones rotate counterclockwise).