There are two types of persimmon: the short fat sweet type known as fuyū-gaki and the taller pepper-shaped astringent (shibui =渋い) persimmon known as hashiya-gaki. In the case of the former, persimmon lovers seem to be split between those who like to eat them hard and crunchy, like an apple, and those who prefer them soft - if you wait long enough, the (by now very sweet) fruit can be scooped out and eaten with a spoon! At the bottom of the page is a video showing how to peel a fuyū-gaki and prepare it for eating.
In Japan a common sight at this time of year are strings of peeled hashiya-gaki hanging out to dry in the sun, which, after a month or so has passed (and daily massages apparently!), become what are known as hoshi-gaki =干し柿 (semi-dried kaki) - the verb hosu (干す) means to air or dry out. Hoshi-gaki, which are shrivelled up, flat, and covered in white sugar powder, are very sweet and often eaten as a snack or even as dessert during winter. Another very popular snack (pictured) are called "kaki no tane" (persimmon seeds), small crunchy long bean-shaped mini rice-crackers (senbei) that actually have no relation to kaki at all!