The Biretta is a hat that moves from the circular hat band into a 4 cornered squarish shape.
The corners are called horns, they may have 3 or 4 horns and the horns can be small like this or quite large and flamboyant.
The hats are worn by priests in the Catholic and Anglican church and also as academics hats in universities, usually denoting someone with a doctorate.
When worn by Bishops or denote ranks the hats can also get extra adornments such as pompoms.
Biretta is also optional head wear for female barristers in England and Wales.
The history of Biretta stretches back at least as far as the 10th Century.