Be all hat and no cattle informal US expression meaning, tend to talk boastfully without acting on one's words.
black hat (or white hat) used in reference to the bad (or good) party in a situation (from the colour of hat traditionally worn by the bad (or good) character in Western films).
hat-trick three successes of the same kind, especially consecutive ones within a limited period, originally referring in cricket to the club presentation of a new hat (or some equivalent) to a bowler taking three wickets successively.
keep something under one's hat keep something a secret.
pass the hat round collect contributions of money from a number of people for a specific purpose.
throw one's hat into the ring express willingness to take up a challenge.