Typically you can find fuyu persimmon in Japanese food markets. They are very common around the christmas/new years holiday here in the west.
They must be very, very, ripe, to the point of looking blemished, before they are ready to eat. At readiness they are meltingly soft and sweet.
An underripe persimmon is astringent and chalky tasting, like a mouthful of raw flour and a chaser of SeaBreez.