Personally, growing up I've usually just eat it raw after it's ripened. That's the only form that i've a experienced a persimmon.
Nonethless, a quick google reveals use as a sauce:
This morning, I cooked the last of the mushy persimmons into a second
batch of persimmon sauce. The first, served over bread pudding on a
whim one evening, was so popular that it was deemed worthy of the last
persimmons. It’s simple and delicious, primarily because, as I
discovered, a generous serving of nutmeg and a bit of meyer lemon is
the best way to season persimmon anything!
Source: Oakland Garden Kitchen
In this article, Oakland Garden Kitchen blogger, also lists a recipes for persimmon bread published by David Lebovitz. Lebovitz has a very comprehensive blog article about persimmons.
Persimmon in Bread
In this blog, he provides an adaptation of James Beard's, "Beard on Bread", Persimmon Bread recipe. If you don't like this version of the recipe. Epicurean has another version.
Persimmon as an Entree
Wine Press NW also once published a recipe of use of persimmon as a glaze for salmon.