Fez

A Fez is a type of hat. The Doctor took one off an exhibit in a museum where the Pandorica was and wore it. He stated that "fezzes are cool", in the same way he often states that "bowties are cool". It was taken from him by Amy Pond and destroyed with a gun by River Song. He was later disappointed to not be wearing a fez anymore, but concluded that he could buy a new one. 

Behind the scenes
Steven Moffat said that Matt Smith grew fond of the fez so they had to destroy it to stop him from insisting on wearing it next series.