There are many different hats - some with special names and historical roots:
The Homburg, a men's hat with a curved brim, became socially acceptable under King Edward VII and Kaiser Wilhelm II. His real breakthrough came from Konrad Adenauer. Previously, only top hats and bowler hats were worn on official occasions.
The pork pie - a hat with a slightly curved narrow brim - also has historical roots. It was made famous in the 1920s by silent film star Buster Keaton. In 1971 it was worn by Gene Hackmann in the film French Connection.