They don't know the exact date in May so let's go with today as Batman's Anniversary! Hitting the stands in 1939 was Bill Finger and Bob Kane's new strip in the pages of Detective Comics #27 The Bat Man! Note: after some research I've discovered this book was on the stands on March 30, 1939-- since I don't have a time machine I can neither post this on the right date or go further back and buy a suitcase full of these comics because they are worth about $2 million a piece now.
Like Superman who came along a year earlier-- it would be some months before Batman became the main figure on the cover of the comic book as National Comics wasn't sure he would be a hit or not.
Bob Kane got most of the credit although we've come to understand over time that while Kane was the one who started the ball rolling it was mostly Bill Finger who came up with all the elements that made the character unique.
Batman didn't have any superpowers and he wouldn't even have an origin until Detective Comics #33 a few months later. In 1940 he was given his own quarterly title in addition to his appearances in Detective Comics and he would get the world's first comic book sidekick with the addition of Robin the Boy Wonder in the pages of Detective Comics #38 in May of 1940.
So Happy 79th Birthday Batman!