To me, persimmons or kaki (柿）are the quintessential fall fruit. Although there are by some counts around 1,000 different varieties of persimmons in Japan alone, they can be broadly divided into two types: sweet (amagaki) and bitter or astringent (shibugaki). The ones in the photo above are the sweet kind, which can be eaten as is as soon as they are ripe. They have firm flesh, and are simply sliced.
The photo below shows a branch laden with the bitter type (shibugaki) hanging outside a house in Naraijuku, a small historic town in Nagano, Japan. These are just decorative, I think, but bitter persimmons are usually hung outside until the bitter substance slowly drips out and the fruit dries out. The resulting dried persimmons (hoshigaki) are as sweet as candy, and the bitter liquid itself is used to lacquer wooden wares in the area. Here are some lacquered bento boxes for sale at a local shop; the dark ones in the back are lacquered with persimmon bitter (kakishibu).