A hat was an article of clothing worn on the top of the head. The Second Doctor wore a stovepipe hat early in his incarnation. The Eleventh Doctor was fond of a variety of hats.
While in theNational Museum, the Eleventh Doctor found a fez which he became very fond of. The fez was later removed by Amy Pond and destroyed by River Song. He sought another fez after this. (DW: The Big Bang, DW: The Impossible Astronaut)
The Eleventh Doctor acquired a Stetson hat from Craig Owens in Closing Time. He was last seen wearing it inside the Teselecta. (DW: The Wedding of River Song).
The Eleventh Doctor owned a top hat, which he wore at Amy and Rory's wedding, after having been poisoned by River (DW: Let's Kill Hitler), and again to conceal a device that allowed him to lure the Squall back into their dimension of origin (NSA: Paradox Lost).