To me, plums are like the last gasp of summer before fall settles in. They are related to other summer stone fruit, like peaches and apricots, but they have a much more elusive flavor.
The kind of plums you are likely to find really depends on where you live. The photo shows a few kinds we can get here. The yellow ones are called Reine-Claude here; in England they're called greengages. I've never seen them in the U.S. They are delicately sweet and very fragrant. They can be made into jam, but to me they are too delicious to have other than just fresh out of hand.
The black ones are called Zwetschge here, and are called damsons I believe elsewhere. When I think of plums, these are the ones that come to mind first. These have a very tart skin, though the flesh below that is sweet. They are the plums most suited for cooking in my opinion. I often make preserves out of them, following the method for apricot preserves . They're also an important ingredient in a dark, strong chutney  that I make every other year or so.
I'm not sure of the name of the pinky-red ones, but they are adorable, with a pointy shape that makes them look like little hearts when cut open.
Finally, there are mirabelles. Mirabelles are miniature plums, each one the size of a large grape. They are incredibly sweet and smell like heaven.
As soon as plums are gone from the market stalls and supermarket aisles, the weather starts to cool down.