There are two types of persimmons currently grown on a commercial scale in the US -- the astringent Hachiya and the non-astringent Fuyu. This primer deals with the Hachiya, which needs to be eaten very, very soft.
From Specialty Produce: When fully ripe the fruits are a beautiful deep orange. Another indicator of ripeness is that the fruit should feel like a water balloon when resting in your hand. The skins are thin, similar to a tomato. The flesh is even deeper orange in color and more striking then the skin. When ripe the flesh is jellied in texture. The flavor is candy sweet and possesses nuances of baking spices, raisins and brown sugar.
From NPR: To eat a Hachiya, remove the calyx (the flower-shaped stem on top) and use a spoon to scoop out the honeyed, custardlike flesh. It's a deliciously messy affair, so have some napkins on hand. Their creamy, sweet flesh makes Hachiyas ideal for baked goods such as muffins, breads and puddings. They also can be pureed and used as a sauce for ice cream or pancakes, or they can be dried and eaten as a snack.
From SF Gate: Place the persimmons in a brown paper bag with an apple or a banana. The ethylene gas apples and bananas produce speeds up the ripening process. Alternatively, keep the fruits out at room temperature in a bowl.
From NPR: Once the fruit reaches its jellylike softness, it can be eaten right away or refrigerated for several days.