"Cyclone" is the generic name for any type of low pressure center that spins counterclockwise (in the Northern Hempisphere). You often hear meteorologists refer to a "cyclonic flow," which basically means a counterclockwise rotation.
Hurricanes, tornadoes, and northern hemisphere typhoons are all "cyclones."
However, hurricanes which form in the Indian Ocean are also called cyclones, while hurricanes that form in the western Pacific above the equator are called typhoons. So, the only difference is the place they form. Confusing?
Why do these weather systems have this cyclonic flow? Because of something called the Coriolis effect.