I'm a zealous persimmon advocate, and am surprised when people see a precariously stacked pyramid in my fruitbowl and ask me what I do with them. As with many weird fruits, I want to respond, "Uh, I eat them–they're delicious as is."
The most common variety (sold at most farmers' markets, at Whole Foods, and in Chinatown) are Fuyu persimmons (above). They're a little squat, resemble shiny mini-pumpkins, and are slightly crisp, like a ripening nectarine. These are also the fruit's safest incarnation: they don't need to be cooked, or even peeled, before eating, and they're sweet as honey.
Hachiya persimmons (pictured at left) are stranger. They're more torpedo-shaped, and if you take a bite before they're fully ripe and pulpy, you'll probably fling the fruit from your hand in disgust–squinting your eyes and scraping your tongue in an attempt to rid your mouth of unimaginable bitterness. That would be the astringent quality most people fear, and the reason (it's silly, really, now that the whole internet thing's caught on) why some eaters avoid persimmons altogether. Hachiyas are ambrosial and decadent when they've reached a state of squishiness that borders on decay. You can eat them with a spoon, or use them for baking.
The other varietal I'm familiar with (and extremely partial to) is a version alien to most people: the Amagaki.
I've been buying them from Twin Peaks Orchard at the Sonoma Valley farmers' market for the past 4 years, and can't find them anywhere else. The seller, Ed, who appeared for a brief window of time at the Crocker Galleria market, says the Japanese restaurants in the city fight for his product, and promises he'll deliver to my doorstep. I'll keep you posted. Anyway, the orchard claims that the Amagaki is actually a Hachiya that (through "an ancient process developed in the Orient") has had the "puckery characteristic" bred out of it. The result is a mouthwatering hybrid of the above varieties: an elongated, acornesque shape; soft and juicy, peachlike (not pulpy) flesh that's edible as soon as you buy it; and a deep, honeyed flavor with a hint of cinnamon. I swear. Try one.