Alright. This will ignore Earth-2/D/other Pre-Crisis Alternate Earths, since that confuses things needlessly. It also ignores Legends of a Dead Earth, DC 1,000,000 and other similar events - just assume those events involve one or more future versions of the characters.
Also, an unquallified reference to ‘Crisis’ refers to Crisis on Infinite Earths. Infinite Crisis will be spelled out.
Batman’s mostly the same character he’s always been - he’s still Bruce Wayne, millionaire playboy by day, ‘Dark Knight Detective’ by night.
However, in the early 90s, his back was broken, by a villain by the name of Bane, in the rather long story-arc Knightfall.
Obviously, that left Bruce paralysed. So a need was left for a new Batman - Jean-Paul Valley, a rather violent religiously motivated assassin is given the job. Eventually, Jean-Paul is shown to be dangerous, and Bruce (whose back has been healed in the meantime) takes the name back. Valley eventually took on the name Azrael, and became an ally (though not a stable one) of the Batfamily.
Soon, he finds he has other things to attend to and hands the cowl over to Dick Grayson (the first Robin, then (and now) Nightwing) for a time. Eventually Bruce comes back and becomes Batman again.
There’s also at least one alternate future (Titans Tomorrow) wherein Tim Drake (the current Robin) becomes Batman after Bruce’s death.
Superman is a little more complicated than Batman, mostly due to events of Crisis on Infinite Earths and Infinite Crisis.
Around the time of Knightfall (I’m blanking on which came first), DC had another big event with Superman - they killed him (temporarily), and replaced him with 4 new heroes - who came to be known as Steel, the Cyborg Superman, the Eradicator, and Superboy. All but Steel claimed to be THE Superman, in a fashion - the cyborg was his body reanimated by technology, Eradicator was his spirit incarnated, and Superboy, his clone. (Those were the claims. The cyborg was lying, Eradicator was wrong, and Superboy…is just complicated.) Obviously, all 3 used the name Superman, at first. Steel actively denied being Superman, although that didn’t stop people from calling him it.
Even after the real Superman came back to life, all 4 remained active - Steel and Superboy as heroes, Eradicator as an anti-hero, the cyborg as a villain.
In Titans Tomorrow, Superboy is shown to have taken the identity of Superman.
Although Superman is again Kal-El of Krypton/Clark Kent, he’s still a tad complicated.
Most non-comics readers would be familiar with him through the older comics, or the George Reeves TV series, or the movies, or Superfriends. The current version of the character is very different than those. (Although Smallville and the Diniverse animated series are relatively close.) He’s got a smaller power set - although the power levels fluctuate, save for his time in his electric form, and a recent return of his Super-Intellect, his power-set has remained more or less constant, and smaller than the absurd shopping list he used to have, since Crisis - and his origin is…changed.
It’s been changed 3 times in the last 20 years.
The first time, John Byrne’s Man of Steel reboot made Krypton a sterile, frankly rather unpleasant place. Visually, it was based, strongly on the Donner movie, but in feel, it was something new. There was also only one type of kryptonite, initially, the green kind, rather than the rainbow of types that existed before.
The biggest changes were to Superman himself - he was depowered, greatly, and instead of having his powers from the time he was a child, as was the case before, they didn’t develop until adulthood. And he was already established as Superman before he found out about Krypton.
This remained more or less the case (with incrimental, and not entirely consistant) changes until Birthright, which pretty much changed the origin to that shown in Smallville. Krypton is a more pleasant place, and Clark starts developing his powers in his late teens.
And now it’s been changed again. It’s not clear how, just yet - again, his powers developed in his late teens, and he spent some time using them in secret (much like Smallville), before finally going public. Krypton has been altered again, but it’s not clear just how, other than some quick glimpses we’ve gotten in Action and Supergirl. The crystaline technology introduced in the movie and Byrne reboot are still around.
The Wonder Woman identity has been used by 5 separate characters within current continuity. 3 main characters, and 2 one-off uses of the name by other characters. The 3 primary ones are Princess Diana of Themiscara, Queen Hippolyta of Themiscara, who took over when Diana was temporarily dead and turned into a goddess, and Artemis, who essentially stole the identity from Diana, who had reclaimed it after she came back to life.
Afte the events of Infinite Crisis, Diana decided to bugger off and travel, to ‘find herself’. Eventually, Donna Troy (the first Wonder Girl) takes up the mantle of Wonder Woman. Several of Diana’s enemies - Dr Psycho, Cheetah, and Giganta, who have been powered up by Circe - kidnap Donna, and eventually the current Wonder Girl, Cassie Sandsmark to draw Diana out of hiding. This was all a plan of Circe’s part to steal Diana’s powers - and identity - because she believed Diana had squandered it by not focussing on protecting and avenging wronged women. How Diana will reclaim the identity, as the above events are from the most recent issue.
In Titans Tomorrow, Cassie has taken the identity of Wonder Woman.
Like Superman, Wonder Woman suffers from some serious reboot issues.
Unfortunately, I’m not exactly a Wonder Woman expert, so there are probably some important details missing from these rundowns:
Pre-Crisis, when Princess Diana came to Man’s World from Paradise Island/Themiscara, she took the name of a woman named Diana Prince (IIRC, Ms Prince was an army nurse, but I’m not 100% on that). She was romantically involved with Air Force Captain Steve Trevor - with the usual complications of superhero romances. She was a founding member of the JLA.
After Crisis, she came to Man’s World well after most of the other superheroes were established, including the JLA (Black Canary was inserted into her role). She did not take the Diana Prince identity, and although she had connections with Steve Trevor, they weren’t romantically linked. She became something of an Ambassador of Peace from Themiscara. Also, Wonder Woman in the JSA (the world’s first superhero team, from the 40s and 50s), due to the Golden Age Wonder Woman being eliminated was filled by a new character named Fury…until John Byrne decided that Hippolyta’s time as Wonder Woman would include time travel and joining the JSA.
After Infinite Crisis, her history is changed again, and, like Superman’s current history, it’s not clear just how it’s been changed. But we’ve been shown some parts - she came to Man’s World early enough to be a founder of the JLA again, although she still didn’t take the Diana Prince identity. Not immediately, in any case - she took it while she was off ‘finding herself’. She was still an ambassador (until the events leading up to Infinite Crisis). Some people read into the first couple issues of the current series that Steve Trevor is back to being her love interest, but I don’t see that, myself. It hasn’t been addressed in the comics, and the creative and editorial staffs at DC are giving conflicting answers, so it’s not clear what’s going on with Hippolyta’s time as the Wonder Woman of the JSA.
I hope that didn’t thoroughly confuse things. <_<