Well, it’s kinda complicated.
The DC Universe started back in the 30’s and 40’s with Superman, Batman, Wonder Woman, the original Green Lantern (Alan Scott), Flash (Jay Garrick), Atom (Al Pratt), Dr. Fate, Wildcat, etc.
After the war, superhero comics lost their popularity and were replaced by war comics, westerns, romances, and so on. Superman, Batman, and a few others survived this period, but most of the heroes from the so-called “Golden Age” were lost to the mists of time.
In the mid-50’s and early 60’s, DC revived their superhero line, in part by reimagning some of their old characters. (The Silver Age) Therefore, there were now a brand-new Green Lantern, Atom, Hawkman, and Flash. In the first issue with the new Flash, it was shown that when police scientist Barry Allen was granted speed powers, he took his name and costume idea from an old Flash comic book starring Jay Garrick, so it was established that this Flash’s stories took place in a world separate from the original’s.
Eventually the two Flashes met up; Barry crossed the dimensional barrier and visited the world where Jay lived. Since Barry’s world was the one in which most DC stories were taking place, it was known as Earth-1. The JSA lived on Earth-2.
There were several differences between Earths 1 and 2. Since their original adventures were tied to WWII, modern stories of Earth-2 showed the JSA as aged. The Earth-2 versions of the Flash, Green Lantern, Atom, and Hawkman were different than their Earth-1 counterparts, because they were the originals which were revamped in the 1950’s. JSA heroes like Dr. Fate or Starman were never revamped, so the Earth-2 versions were the only ones. (Conversely, heroes like Metamorpho were new in the Silver Age so they existed only on Earth-1). Superman and Batman (and some others) had strips that lasted through the interregnum, so they existed in both Earth-2 and Earth-1 versions. (But, again, since the Earth-2 versions lived through WWII, and the modern versions showed up “about 10 years ago,” the E-2 versions were much older.)
The multiple earths concept proved a very popular one, especially when the Earth-1 JLA/Earth-2 JSA meeting became an annual event. Their first team-up (which I haven’t read, so my details may be a bit off) came when the JSA contacted the JLA for assistance, and the story was titled “Crisis on Earth-2.” Next year, the JLA and JSA went to another dimension, Earth-3. Later, they visited a dimension populated by the characters originally published by the Fawcett company in the Golden Age (Captain Marvel, Mary Marvel, Black Adam), which DC had purchased the rights to. DC also purchased the rights to the old Quality characters (Phantom Lady, Uncle Sam) and Charlton’s, too (Blue Beetle, the Question). Each of these groups lived on their own planet and had very little commerce with the others. (The first several of these JLA-JSA team-ups, all of which were titled “Crisis on Earth-whatever” are being reprinted in a new TPB in a few months called, IIRC, Crisis on Multiple Earths.)
By the 70’s, there were some series that took place on Earth-2, such as a revival of the JSA’s feature in All-Star comics; America vs. the Justice Society was another. (Captain Carrot and the Zoo-Crew took place on Earth-C, which was visited by Superman!) The series you read told of the adventures of Earth-2’s JSA. It’s not just that the Flash’s costume was different; he was a different guy! It was the Earth-2 Batman that had passed on. The “real” Batman, inhabitant of Earth-1, was seen every month in his own titles.
Eventually it was decided that this was all too confusing, what with the JLA, JSA, Charlton heroes, Shazam! heroes, Quality heroes, and the Crime Syndicate of America all living on different worlds. The decision was made to consolidate all the DC-owned stories into a single dimension to the extent possible. This was done via a huge crossover. Since the original stories were all “Crisis on Earth-whatever,” this was called the “Crisis on Infinite Earths.”
During Crisis, a bad dude named the Anti-Monitor destroyed pretty much all of the multiverse, obliterating infinite worlds. The heroes of all the multiverse managed to save six earths, but they were unstable, so they were merged into a single earth. After the dust settled, what happened was this. Most of the original Golden Age comics (whether published by DC or Quality) really happened back around the time of WWII. The Phantom Lady was in the JSA back then, as was Starman. The Charlton, Fawcett, and modern DC heroes all existed in the present day, having appeared there “around ten years ago.” That’s why in today’s JSA comic, Jay Garrick, the original Flash, is an old man. (He should be older, but he was irradiated by time particles or something.)
Of course, there are some problems, particularly with heroes that lasted through the interregnum. For instance, post-Crisis, only the Earth-1 Batman ever existed, not the Earth-2 version who fathered a child named Helena Wayne, aka the Huntress. So who is the Huntress now? (It turns out, she’s Helena Bertenelli, daughter of a mob family.) What’s the deal with Black Canary, an original JSA heroine who crossed over to live with the JLA pre-Crisis on Earth-1? Now, there are two. The JSA version and her daughter, the modern version. (Kinda the same thing happened to Wonder Woman.) And given that the Green Arrow now never existed in the Golden Age, who the hell was in the Seven Soldiers of Victory, a team in which he was included back in the Golden Age?
These questions are answered as they go along, but they’re not always answered well, which causes additional problems.
But anyway, I hope that answered your questions.