Jennifer, you are talking about Asian persimmons. Native persimmons are about the size of a ping-pong ball, and very mooshy when ripe. They have 4 or 5 large seeds. They grow throughout the southeast.
When they are green, they are very astringent, and will just about turn your mouth inside out.
Some folks predict the weather by splitting the hard seeds lengthwide. There is a white structure inside the seed, and if it looks like a knife, the winter will be "cutting cold", and a spoon means lots of snow.
I don't care for them much, but Mitchell Indiana has a Persimmon Festival, with prizes for the best persimmon pudding.
I just haven't been the same
since that house fell on my sister.