Similar hats with a C-crown (with an indentation for the head in the top of the crown) are occasionally called fedoras. The brim goes all the way around, and often there will be a hat band as well. A trilby hat is somewhat similar to a fedora, but typically has a shorter brim, and the back of the brim is distinctively more sharply upturned as a result.
The early 1950s switch from large lapels and ties to thin ones, resulted in shorter-brimmed hats, and this likely played a role in the fedora eventually being deemed a non-essential item. Also playing a part was the shrinking automobiles of the mid-1950s, which often made it difficult to wear a hat while driving. By the early 1970s, the fedora was seen as a dead fashion, typically only worn by older and/or more traditional men.
Source: Wikipedia > Fedora
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