Hangers are purely for decorative use and serve no functional purpose. They are often used to hang signs from, but many other decorative uses are possible.
You can change the direction of the hanger by facing your avatar left or right before placing the hanger.
Early in the game, hangers were one of the side-facing items that did not line up correctly when facing to the left. While the left facing hanger was one of the more subtle decorations, one could not align a chain with the hanger.