Pistons, referring to organs, not internal-combustion engines, are the little (usually white) buttons between the manuals that change stop combinations. There is a "set" piston to put a given combination on a given piston, thus they are known as "adjustable" pistons. There are also toe studs that duplicate general and pedal pistons.
But this is in the U.S., and there are differences in other countries. It seems that in Europe, especially Germany, many organs have only "general" pistons under the first manual. And I understand they are not always "adjustable".
Yes, when a piston is pressed, the stop knobs or tabs and couplers, etc. move to the combination set on that piston. That way, the organist can also add or subtract stops by hand. There are usually (in the U. S.) pistons for each manual (division) plus "general" ones that change all the divisions at once. Also there are "reversible" pistons, usually to control couplers and 32-foot stops. Those are not changeable and have just one particular purpose.
All this came about around the late 19th or early 20th centuries when electrical action became common. Nowadays some builders provide new tracker-actions organs with electrical stop control, so they can have the convenience of pistons.
Depending on the size and complexity of a particular organ, there are sometimes pistons to do certain things, such as kill off the reed stops, etc. I suppose at least theoretically, they could be used to do anything the builder wants them to do, such as tell the preacher it's time to shut up.
I hope this clarifies what "pistons" are!!