A hat style characterized by a creased crown and a snap brim is a fedora.
A fedora is generally a soft felt hat with an indented or pinched crown and a brim that can be shaped, often turned down at the front and up at the back. Its crown may have a center crease, a teardrop shape, or another molded indentation. The snap brim is especially associated with the classic fedora silhouette, although individual examples vary.
The word “fedora” dates to 1891 and came from a play titled Fedora, whose heroine wore a similar soft, creased hat. The style became especially popular in the early twentieth century and gained further visibility after Edward, Prince of Wales, adopted it in the 1920s.
Fedoras are often confused with trilbies and homburgs. A trilby usually has a narrower brim and a sharper, more sharply angled crown, while a homburg traditionally has a stiffer structure and a pronounced single dent. Modern fashion often uses “fedora” loosely, but the creased crown and flexible brim remain its defining clues.