Headgear

Headgear was any article of clothing worn on a person's head.

Bajoran hats

Bajoran vedeks and kais, such as Winn Adami, wore special types of hats. (Star Trek: Deep Space Nine)

Vedek Winn's hat was designed by Robert Blackman. He created it as an "interesting and curious shape that viewers haven't seen. It's about rhythm and other things." Additionally, he admitted that the style resembles the Sydney Opera House, though he remarked, "I didn't sit down and say, 'Oh, the Sydney Opera House – let's make a hat out of it!'" However, Blackman did accept that the likeness was intentional and subconscious, as well as it being largely accidental. (Star Trek: Deep Space Nine Companion (p. 69))
Winn actress Louise Fletcher once referred to the hat she wore when Winn was a kai as "a Robin Hood-style hat." (Star Trek Monthly issue 34, p. 50)

Headdress

A headdress was a piece of material worn around the back of a head. It was normally seen being worn by Ferengi, and as part of the Ferengi uniform.

The recreated Ferengi sciences officer aboard Terrasphere 8 was wearing a headdress matching his uniform color. (VOY: "In the Flesh")

A turban was a type of headdress worn by some Humans in the 20th century, as well as in the 2380s. (TOS: "Space Seed"; LD: "Second Contact")