A Trilby hat is kind of the little brother of a standard fedora hat. Like the fedora, a trilby hat also has a curled tapered crown with a front pinch, but is slightly more streamlined with a shorter snap brim, which usually has a simple stitched brim and sometimes a very narrow band across the base of the crown.
Originally primarily associated with the trench coat, the trilby hat evolved in the 1920s into a softer hat that was often worn in the English countryside and also on the racetrack. For these reasons, it is typically seen as a slightly more informal hat than the standard fedora, but not by much.
Trilby hats also look particularly good on men with strong facial features, so take that into account when choosing a hat!