Tropical cyclones of the North Atlantic Ocean, from 1950 to 1952, were identified by the phonetic alphabet (Able-Baker-Charlie-etc.), but in 1953 the US Weather Bureau switched to women's names.
From there the rest of the world eventually caught on and the naming rights were given to the World Meteorological Organization.
The WMO uses different sets of names depending on which part of the world the storm is in.
Around the U.S., only women's names were used until 1979, when it was decided that they should alternate a list that included men's names too.
There are 6 different name lists that alternates each year.
If a hurricane does significant damage like Hurricane Katrina, the name is retired and replaced with another.