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The fez is a red short cylinder with a tassel on the top of it. It was popularized during Ottoman period, but still is an icon of Ottoman culture. Fez was used in the Middle East, Eastern Europe, and North Africa, so it's unclear where were its origins. It was a regular part of uniforms of many countries and today is mostly exotic fashion item and part of national costumes.
The baseball cap is a soft hat with a peak and traditionally was part of baseball uniform. This cap today even symbolizes American Culture. The first version was known as "Brooklyn style" cap. During the time the peak was longer - to protect eyes from the sun - so it's logical that today everyone wears it: truck drivers, tennis players, and police officers.
Ushanka is Russian hat with round crown and ear flaps, usually made of fur. This hat is one of the symbols of the Soviet Union because it was developed and popularized in Russia during Civil War. Today ushanka is standard winter military hat in many countries, but also a fashioned hat made of different materials, not just fur.
Sombrero is a Mexican hat famous for its very wide brim, high and pointed crowns and a slightly upturned edge. Over the last several centuries, sombrero and poncho managed to become the iconic part of the Mexican society. Today this hat is worn not only in Mexico but also all over the world.