Here's a list of the most common .32 revolver rounds you'll encounter...
.32 S&W - Once hugely popular because of the small handguns that chambered it starting in the 1870s. Obsolete, but ammunition is still manufactured.
.32 S&W Long - A lengthened version of the .32 S&W. Also once very popular. Still fairly commonly found, chambered in revolvers until fairly recently. Ammo still commonly available. Also chambered in some high-end European semi-auto target pistols.
.32 Short and Long Colt - Competitors to S&W's two rounds. NOT interchangeable with the S&W rounds. Not nearly as popular, and obsolete for a long time, although you'll still find Short Colt ammo.
.32 H&R Magnum - A lengthened, more powerful version of the .32 Long. Never particularly popular, but still encountered.
.327 Federal - Longer than the H&R and more powerful. Designed to give a 6 shot compact revolver with power approaching the .357 Magnum. Neat concept, but appears to be dying.
.32-20 - Originally a Winchester rifle round, became a VERY popular Colt & S&W revolver round. Only handgun I know that still chambers it is the T/C single shot.
There were some other .32 rounds manufactured/chambered in revolvers over the years, such as the S&W .32-44 target and the .32 Merwin & Hulbert, but they are LONG obsolete, guns are not commonly encountered, and the ammo is really in the collectors' realm.
There's also the .320 Revolver, designed in Britain, and which served as the inspiration for the .32 Short Colt. There were a lot of European revolvers chambered for this cartridge, so runs are occasionally produced in Europe and South America.