No one wears them anymore, but at one point in Korean history, these hats were a common sight :
The gat (갓) was a hat worn by upper class Korean men. This video is a clip from the Korean drama called Yisan (이산). If you look at the first seconds of the video, you can see how the gat marked a person’s status. The servant has a plain black sash tying the gat to his head, but the king’s gat has a sash strung with expensive beads.
A 정자관 was a hat worn by upper class Korean men. The 갓 was worn as a part of formal attire, but the 정자관 was worn as a part of casual attire. The 정자관 was made of horse hair and men wore these hats to denote their status, but also to protect a type of hairstyle called a sangtu (상투).
A 호건 was a long pointy hat worn by upper class unmarried boys. You can still see a variation of this type of hat worn in 돌 ceremonies. In the video, the length of the hat has been shortened at the back.