The 26 inch "lightweight" bike sizes:
599mm="American"=26x1.375
597mm=650=26x1-1/4=26x1-1/4x1-3/8
590mm=650A=26x1-3/8
584mm=650B=26x1-1/2
The easiest way to remember them, I think, is to use the bead seat diameter in MM measurement. That measurement gives you an idea as to how close together the tire fittings are, and whether they will work. Using the nominal/imperial size can be very confusing.
Old American lightweights used the peculiar 599mm "American" size. The tires usually say 26x1.375. This is the size most common on pre-war lightweights (including Schwinn). Schwinn moved to 597mm after WWII. The other manufacturers stuck with 599mm after the war initially but eventually went to 590mm so their bikes could use the common English import tires. These tires are not made any more. Fortunately, many (though not all) 599mm rims will also accept 597mm tires. Some even have a stamp inside the rim saying they will work with both sizes.
The 597mm size is called "650" in the French system. This size was known in England as 26x1-1/4 and in the US additionally as 26x1-1/4x1-3/8. In England, this was a popular size for "clubman" type bikes and similar sportier lightweights. In the US, Schwinn adopted this size for its post-war rims. Tires came in basically two sizes, a club tire (32-ish mm) and a touring tire (37-ish mm). The 597mm tires will often work on 599mm rims with some persuasion. The 597mm will NOT work on 590mm rims. The Kenda K23 37-597 is the only commonly available new tire in the US today. There is a bike shop in England selling 32-597 club tires as well on a smaller scale.
The 590mm size is the size used by most English 3-speeds in the USA. This is the common size that the Raleigh Sports and similar bikes used. The later American-made 3-speeds from Columbia/Westfield use this size as well. Tires are still widely available. The Schwalbe, Panaracer, and Michelin tires are worth a look if you want something better the Kendas or Duros in this size. Widths on the tires vary quite bit. The Schwalbe and Panaracer tires tend to run wider. You cannot use these tires on the Schwinn 597mm rims or the old British clubman rims.
The 584mm is the French 650b "randonneur" size. It was also used as the tire for some of the rod brake British roadsters that had 26-inch wheels. Some American bikes did use this size - tandems, welterweights, etc. It will not fit on any of the other rims sizes above. It was a dying size for awhile, but has had somewhat of a renaissance. There are several tires available, including some very good ones from Japan.