Fez

A Fez was a type of hat. The Doctor took one off an exhibit in a museum where the Pandorica was and wore it. He stated that "Fezs 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. 

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.