1.
hat shop; hatter
A shop specializing in selling hats, or a person who makes or sells hats.
A compound of 帽子 (hat) and 屋 (shop, dealer). The suffix 屋 is one of the most productive word-forming elements in Japanese, attaching to products to create shop names: 花屋 (flower shop), 本屋 (bookshop), 魚屋 (fish shop), etc.
帽子屋 can also refer to the person who runs the shop (the hatter), following the same pattern as 八百屋 (greengrocer, or the greengrocery).