Who's the Earth-2 Batman?

Who's This? You mean Catwoman's husband?

The facts: Obviously, everyone knows Batman, but what must be understood here is that when DC decided its Golden Age characters' adventures took place on a separate Earth, it meant the Batman who lived through World War II could be allowed to get older on that Earth, in a way "current" Batman never could. It allowed writers to imagine a future for Bruce Wayne where he mostly retired from being the Bat, became Commissioner, and married Catwoman. They had a child who became the Huntress. A happy ending for the Bat? It's certainly rosier than The Dark Knight Returns or even Batman Beyond. So while the Batman of Detective Comics #27 is THIS Batman, the first appearance of an "Earth-2 Batman" is really Justice League of America #82 by Denny O'Neil and Dick Dillin. The character would later appear in The Brave and the Bold, All-Star Squadron, America vs. the JSA (where his diary is used against the heroes after his death, which occurs in 1979's Adventure Comics #462, see below), Secret Origins, etc. but the Crisis effectively erased him from the single-Earth timeline.
How you could have heard of him: There have been permutations of Earth-2 Batman since the return of a Multiverse, but nothing major. Notably, the New52 Earth-2 series made him Thomas Wayne (Flashpoint has a lot to answer for), kills him off, and then makes Earth-2 Dick Grayson the new Batman.
Example story: Adventure Comics #462 (March-April 1979) "Only Legends Live Forever" by Paul Levitz and Joe Staton

It's the late 70s and All-Star Comics has been cancelled, the contemporary JSA moving house to the 68-page Adventure Comics. In the second issue of that run, Batman dies, and you can do that with a redundant character living on another Earth, his legacy is well assured by Robin and Huntress, and he's been retired since the death of his wife.

So what pulls him back into action one last time? There's this convicted criminal, Bill Jensen, who claims Bruce Wayne framed him to become Police Commissioner. One morning, he woke up with cosmic-level powers and escaped to take revenge. Dude is powerful enough to defeat the entire JSA (including a powerhouse like Dr. Fate), so can Batman be much help? At least he hasn't completely lost his skills.

Jensen is surprised to see Batman, who disappeared months ago, but by golly, the Guardian of Gotham is gonna bring him Bruce Wayne or people are gonna be made to suffer. The JSA awakens, strike, but are once again all knocked out. What can Batman do?! What he's always done. Persevere. Not give up. Struggle ever forward. And when Jensen recognizes the Batman IS Bruce Wayne, he explodes with anger and satisfies whatever cosmic force inhabited him...

...and takes the Batman with him, before his daughter's eyes.

EPILOGUE: Funeral for a Bat

A well-attended funeral for a legend. He is buried next to Selina, and close to his parents. Helena Wayne weeps uncontrollably. Dick Grayson sheds a tear. A promise is made... ALMOST!

If it weren't such a sad moment, it would be comical. Robin doing the whole legacy speech and Huntress stopping him like that. But she tells him there could only be one Batman and he's dead. We carry on as Robin and Huntress, but...

Oh boy, that's creepy. I wonder what Helena would think of what would happen on post-Crisis Earth on that score. Maybe she'd be too busy wondering what the heck happened to HER instead. As for what possessed Jensen, the JSA will have to keep digging. Now is the time for grieving.

Who's Next? The other one.