That's the Postman's Hat, isn't it? That is a fine thing. It is filled with the joy of freedom.
The Postman's Hat is given to Link by the Postman on the Final Day, who is distressed about wanting to flee Clock Town. After Link shows him the priority mail, he will pull himself together and quickly deliver Kafei's letter to Madame Aroma. The Postman will then meet up with Link outside the Milk Bar and give him the Postman's Hat, telling him that he has decided to flee Clock Town. The Postman will then run off, leaving Link to his important deliveries.
Link can use the authority that comes with the mask to check inside mailboxes. The first time, Link will find a Piece of Heart. Every subsequent time, he will find a Green Rupee.