A cyclone is the general name referring to a storm (technically called a ‘weather system’) which has strongly rotating air inside.
In the Pacific north-east and in the Atlantic/Caribbean, tropical cyclones are known as hurricanes.
In the Pacific north-west, severe tropical cyclones, with maximum mean winds above 117 km/h, are called typhoons.
Various types of cyclone occur in the atmosphere. These include tropical cyclones (occuring in the low latitudes), extratropical or subtropical cyclones (occuring in the midlatitudes) and some general low-pressure systems (eg, the famous east coast low of Australia).
Tropical cyclones have a specific structure in terms of winds and temperature inside the storm, which distinguish them from cyclones in the other geographic regions.
Tropical cyclones develop over most of the oceans worldwide, except for the eastern South Pacific and the South Atlantic.
A hurricane is the specific name given to tropical cyclones over North Atlantic. So hurricanes have the same physical structure as all the other tropical cyclones.
The key difference between a hurricane and a cyclone is their origin: hurricanes form because of the hot air propagating as waves from the African Sahel region, while many other tropical cyclones in the Pacific and Indian Ocean form from local summer monsoon winds and moisture. This explains why tropical cyclones almost never occur in winter.
Weather forecasters worldwide assign different categories of severity to their hurricanes and cyclones. Examples are the categories 1-5 used in the North American region, and the names typhoon and supertyphoon used in East Asia.
In the Australian region, we simply call the storms tropical cyclones and also have our own categories of cyclone intensity that refer to the maximum wind speed found inside the storm.