Cyclonic as a direction of flow is counterclockwise in the Northern Hemisphere, and clockwise in the Southern Hemisphere. It is the opposite of anticyclonic.
The Coriolis Effect is responsible for the difference in direction in the two hemispheres. In the Northern Hemisphere, winds curve to the right, so as air moves towards the low pressure at the center of a storm, it bends to the right and creates an overall counterclockwise rotation.
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This image is exaggerated to show the origin of the spin. Air still reaches the center of the system.
Cyclones (a generic term encompassing hurricanes, tropical storms, and typhoons) are always cyclonic, due to their low-pressure nature. High pressure systems, in contrast, are always anticyclonic. As a notable exception, tornados have been known to form in both cyclonic and anticyclonic configurations.
Node what you don't know: definition from Wikipedia. Google concurs and provided some further fact-checking and research.