A hat box is a container used to store your hat for transportation and storage. They can be made of a variety of materials: plastic, cardboard, metal, leather. These containers may or may not be padded. They may also have carrying straps. The purpose of a hat box is to prevent the hat from getting dusty, or from getting damaged.
Of course, there are more purposes to hat boxes than just protecting a hat. These can include: photos, letters, jewelry, and fabric storage. Due to their large size, hat boxes are very accommodating.
The shown hat box can be found at: https://shop.nordstrom.com/s/calpak-baye-small-hardcase-hat-box/4633737?country=US¤cy=USD&cm_mmc=google-_-shopping_ret-_-645528200-_-9382829273_3710e56f-3b28-46a6-8b91-6a9eb001eeb5&cm_mmca1=pla-69992884673_10343095&gclid=EAIaIQobChMImsH2ytXH2wIVCgtpCh3YaQlTEAQYCiABEgJZRfD_BwE