This red hat is a hat worn by members of the modern Freemasons
It was given as a gift by a friend who is a member, I have no idea what rank or role the hat is for, if indeed you need a rank to wear it.
The hat is basically a felt covered stiffened ring of card or thin plastic with felt stretch over the crown, it's very lightweight but well made and still feels sturdy.
Freemasons are an organization that originated in medieval times, as a body that controlled and looked after local stonemason and represented them with the local bureaucracy. Nowadays they are a ritualised group that helps businessmen network and do charity work.
They are often associated with a degree of conspiracy due to there requirement that each rank keeps the knowledge of what they learn as they progress secret from the ranks below.